A proof of A. Gabrielov's rank Theorem
aa r X i v : . [ m a t h . C V ] A ug A PROOF OF A. GABRIELOV’S RANK THEOREM
ANDRÉ BELOTTO DA SILVA, OCTAVE CURMI, AND GUILLAUME ROND
Abstract.
This article contains a complete proof of Gabrielov’s rank Theo-rem, a fundamental result in the study of analytic map germs. Inspired bythe works of Gabrielov and Tougeron, we develop formal-geometric techniqueswhich clarify the difficult parts of the original proof. These techniques are ofindependent interest, and we illustrate this by adding a new (very short) proofof the Abhyankar-Jung Theorem. We include, furthermore, new extensionsof the rank Theorem (concerning the Zariski main Theorem and eliminationtheory) to commutative algebra. Introduction
This article contains a complete and self-contained proof of Gabrielov’s rankTheorem, a fundamental result in the study of analytic map germs. Let us brieflypresent its context and the theorem.Let ϕ : ( K n , −→ ( K m ,
0) be an analytic map germ of generic rank r overthe field K of real or complex numbers, that is, the image of ϕ is generically asubmanifold of K m of dimension r . When ϕ is algebraic, by a theorem of Chevalley[Ch43] (in the complex case) and Tarski [Ta48] (in the real case), the image of ϕ isa constructible set, that is, a set defined by polynomial equalities and inequalities.In particular the Zariski closure of the image has dimension r . If ϕ is complexanalytic and proper, Remmert proved that the image of ϕ is always analytic [Re57].In the case of an analytic map germ, however, the image is very far from beinganalytic. For instance, Osgood gave in [Os16] an example for which the dimensionof the smallest germ of analytic set containing the image is greater than r and,subsequently, Abhyankar generalized this example in a systematic way, see [Ab64].In this context Grothendieck, in [Gro60], asked if the dimension of the smallest germof analytic set containing the image (the analytic rank ) is equal to the dimension ofthe smallest germ of formal set containing the image (the formal rank ). Gabrielovanswered negatively to this question [Ga71], and provided a sufficient condition forthe answer to be positive [Ga73]. Roughly speaking, the result is the following (seeTheorems 1.2 and 1.4 for a precise formulation): Gabrielov’s rank Theorem.
For a K -analytic map ϕ : ( K n , −→ ( K m , , ifthe generic rank of ϕ equals its formal rank, then it also equals its analytic rank. Gabrielov’s rank Theorem is a fundamental result because it provides a simplecriteria for regular maps, that is, maps whose three ranks coincide at every point
Mathematics Subject Classification.
Primary 13J05, 32B05; Secondary 12J10, 13A18,13B35, 14B05, 14B20, 30C10, 32A22, 32S45.The third author would like to thank J. M. Aroca, E. Bierstone, F. Cano, F. Castro, M. Hickeland M. Spivakovsky for the fruitful discussions he had about this problem. of their sources. On the one hand, regular analytic maps constitute an impor-tant subclass of analytic maps, which share basic properties with (Nash) algebraicmaps. For example, the images of regular proper real-analytic mappings form aninteresting subclass of closed subanalytic sets, whose study goes back to works ofBierstone, Milman and Schwartz [BM82, BS83]; see also [BM87a, BM87b, Pa90,Pa92, BM00, ABM08]. On the other hand, non-regular analytic maps are at thesource of several pathological examples in complex and real-analytic geometry, e.g.[Os16, Ab64, Ga71, Pa89, BP18, BdSB19].Nevertheless, the original proof of Gabrielov is considered very difficult, c.f. [Iz89,page 1]. For example, in the 70’s and 80’s, several authors studied analytic mapgerms via more elementary techniques, avoiding Gabrielov’s rank Theorem [MT76,EH77, Mal77, BZ79, CM82, Iz86, Iz89], sometimes re-proving weaker versions of it.In the applications to calculus of variations [Ta81] and foliation theory [CM82] (c.f.[CCD13]), Gabrielov’s rank Theorem is cited but the authors prefer adding furtherarguments in order to use a Frobenius type result of Malgrange [Mal77] instead.Moreover, some specialists believe that the proof contains ideas which would leadto the development of important new techniques concerning formal power series.In [To90], Tougeron proposed a new proof of Gabrielov’s rank Theorem which,unfortunately, is still considered very difficult (and contains some unclear passagesto us, which we point out in the body of the paper).The first and main goal of this paper is to present a complete proof of Gabrielov’srank Theorem. We have been strongly influenced by the original papers of Gabrielov[Ga73] and Tougeron [To90], but we do not fully understand either one of theirproofs. We provide, therefore, several new arguments. The initial part of the proof,given in § §§ § § §§ ABRIELOV’S RANK THEOREM 3
Given the history of the rank Theorem, we have made an extra effort to makethe paper as self-contained as possible. We rely only on well-known results ofcommutative algebra, complex geometry and analysis which can either be found inbooks (e.g. resolution of singularities, Artin approximation) or admit simple proofs(e.g. Abhyankar-Moh reduction theorem [AM70]). All other necessary results havebeen revisited.1.1.
Gabrielov’s rank Theorem.
Let K denote the field of real or complex num-bers. We denote by x = ( x , . . . , x n ) and u = ( u , . . . , u m ) two vectors of indeter-minates. The ring of convergent (resp. formal) power series in n indeterminatesover K will be denoted by K { x } (resp. K J x K ). A morphism of convergent powerseries rings is a ring morphism of the form ϕ : K { x } −→ K { u } f ( x ) ϕ ( f ) := f ( ϕ ( u ) , . . . , ϕ n ( u ))where the ϕ i ( u ) ∈ K { u } for i = 1, . . . , n , do not depend on f . Note that ϕ inducesan analytic map germ between smooth analytic space germs, that is: ϕ a : ( K m , −→ ( K n , u ϕ ( u ) := ( ϕ ( u ) , . . . , ϕ n ( u ))where ϕ a is the geometrical counter-part of ϕ . We are ready to provide a precisenotion of ranks: Definition 1.1 (Ranks: the smooth case) . Let ϕ : K { x } −→ K { u } be a morphismof convergent power series rings, and denote by b ϕ : K J x K −→ K J u K its extension tothe completion. We definethe Generic rank: r( ϕ ) := rankFrac ( K { u } ) (Jac( ϕ a )) , the Formal rank: r F ( ϕ ) := dim K (cid:18) K J x K Ker( b ϕ ) (cid:19) = n − ht(Ker( b ϕ )) , and the Analytic rank: r A ( ϕ ) := dim K (cid:18) K { x } Ker( ϕ ) (cid:19) = n − ht(Ker( ϕ ))of ϕ , where Jac( ϕ a ) denotes the Jacobian matrix associated to ϕ a , rank Frac( K { u } ) ( M )denotes the rank of the matrix M over the field of fractions of K { u } , and dim K ( A )denotes the Krull dimension of the ring A .We can interpret geometrically the three ranks of ϕ via its geometrical coun-terpart ϕ a : ( K m , −→ ( K n ,
0) as follows: for every sufficiently small open set V ⊂ K m containing the origin, the image ϕ a ( U ) is a subset of K n which containsthe origin. It induces, therefore, a set germ ( Z, ⊂ ( K n , ϕ ) is thedimension of ( Z,
0) at a generic point, r F ( ϕ ) is the dimension of the formal Zariskiclosure of ( Z,
0) in ( K n , A ( ϕ ) is the dimension of the analytic Zariski closureof ( Z,
0) in ( K n , ϕ ) r F ( ϕ ) r A ( ϕ ) (see e.g. [Iz89]) . Gabrielov’s rank Theorem provides a simple criteria to show that all of the ranksare equal:
Theorem 1.2 (Gabrielov’s rank Theorem: the smooth case) . For a morphism ofconvergent power series rings ϕ : K { x } −→ K { u } : r( ϕ ) = r F ( ϕ ) = ⇒ r F ( ϕ ) = r A ( ϕ ) . A. BELOTTO DA SILVA, O. CURMI, AND G. ROND
We are now interested in investigating singular spaces. Just as in [Iz89], theessential case to consider is the complex-analytic one, and we specialize our studyto K = C (see Remark 1.5(2) below for a discussion on the real-analytic case). An analytic C -algebra A is a local ring of the form A = C { x } /I where I is an idealof C { x } . A morphism of C -analytic algebras is a morphism ϕ : A −→ B where A = C { x } /I , B = C { u } /J , and ϕ is induced by a morphism of convergent powerseries rings C { x } −→ C { u } . We denote by b ϕ : b A −→ b B the map induced by ϕ between the completions of A and B . It is called the completion morphism of ϕ .Note that the definitions of the formal and analytic ranks of ϕ easily extend toa morphism of C -analytic algebras ϕ : A −→ B . The generic rank, nevertheless,does not extend in a trivial way because we can not define the Jacobian in thesingular context. In order to define the generic rank, note that ϕ : A −→ B also induces a morphism between (not necessarily smooth) analytic space germs ϕ a : ( Y, b ) −→ ( X, a ) so that A = O X, a and B = O Y, b , where O X, a and O Y, b denotethe local rings of analytic function germs at a and b respectively. We suppose that B is an integral domain, which is equivalent to ( Y, b ) being irreducible and reduced.Denote once again by ( Z, a ) ⊂ ( X, a ) the germ image of ( Y, b ) by ϕ a . We define the generic rank of ϕ a as the generic dimension (over C ) of ( Z, a ), and we denote it byr( ϕ a ). It coincides with the generic rank (given by definition 1.1) of ϕ a restrictedto Y r Sing( Y ). Definition 1.3 (Ranks: the general case) . Let ϕ : A −→ B be a morphism of C -analytic algebras where B is an integral domain, and denote by b ϕ : b A −→ b B itsextension to the completion. We definethe Generic rank: r( ϕ ) := r( ϕ a ) , the Formal rank: r F ( ϕ ) := dim b A Ker( b ϕ ) ! = dim( b A ) − ht(Ker( b ϕ )) , and the Analytic rank: r A ( ϕ ) := dim (cid:18) A Ker( ϕ ) (cid:19) = dim( A ) − ht(Ker( ϕ ))of ϕ . We recall that dim( A ) = dim( b A ) when A is a Noetherian local ring.We note that the inequalities (1) are again valid in this context. We are readyto formulate the general version of Gabrielov’s rank Theorem: Theorem 1.4 (Gabrielov’s rank Theorem) . Let ϕ : A −→ B be a C -analyticmorphism, where B is an integral domain. r( ϕ ) = r F ( ϕ ) = ⇒ r F ( ϕ ) = r A ( ϕ ) . Remark 1.5 (On the hypothesis of Theorem 1.4) . (1) (The complex-analytic case) The Theorem holds true if B is reduced (thatis, free of nilpotent elements) instead of an integral domain. This can bededuced from Theorem 1.4 (see, e.g. [Ga73, Proposition 5.6]).(2) (The real-analytic case) Given a real-analytic morphism ϕ : A −→ B ,there exists a well-defined complexification ϕ C : A C −→ B C . Following[Iz89, § ϕ as the ranks of ϕ C . It is nowstraightforward to prove that Gabrielov’s rank Theorem holds whenever B C = B ⊗ C is reduced. This statement, nevertheless, hides two difficultiesin working in the real case which can be illustrated via the integral domain ABRIELOV’S RANK THEOREM 5 B = R { x , x } / ( x + x ). First, note that its complexification is not anintegral domain. Secondly, if we denote by ( Y, b ) the (real) geometricalcounterpart of B , note that Sing( Y ) = Y . It follows that the generic rank,as defined above, is intrinsically complex and does not coincide with thegeneric dimension of the image of Z = ϕ a ( Y ). Both of these issues can bedealt with, in an easy way, by supposing that B is a real-closed integraldomain (that is, B = R { x } /I where I is a real-closed prime ideal). Thiscondition guarantees that B C is an integral domain and that the genericrank coincides with the generic dimension of the image Z = ϕ a ( Y ); inparticular Sing( Y ) = Y .In the rest of the paper, we focus on the essential case K = C .1.2. Applications and variations.
Strongly injective morphisms.
The problem raised by Grothendieck has beengeneralized to the following problem: given a morphism of C -analytic algebras ϕ : A −→ B , when does b ϕ ( b A ) ∩ B = ϕ ( A ) hold true? If the equality is verified, wesay that ϕ is strongly injective . This terminology was introduced by Abhyankar andvan der Put [AP70] who were the first ones to investigate this question. In particularthey proved that ϕ is always strongly injective when A is a ring of convergent powerseries in two variables over any valued field.Without this assumption on the dimension, the equality b ϕ ( b A ) ∩ B = ϕ ( A ) doesnot hold in general (see Example 1.15 and (4)). In this work, we provide a simpleproof of the following characterization: Theorem 1.6 ([Ga73, Theorem 5.5],[Iz89, Theorem 1] c.f. [EH77]) . Let ϕ : A −→ B be a morphism of analytic C -algebras where B is an integral domain. Then r( ϕ ) = r F ( ϕ ) = r A ( ϕ ) ⇐⇒ ϕ is strongly injective . The direct implication of this Theorem has first been proven by Gabrielov [Ga73,Theorem 5.5]. Eakin and Harris [EH77] also gave a proof of this implication (avoid-ing Gabrielov’s Theorem 1.4), in the case where A and B are rings of convergentpower series. They also proved the reverse implication in the same situation. Fi-nally Izumi [Iz89] gave a proof of the equivalence (avoiding Gabrielov Theorem 1.4)in the general case. In §§ Variations of Gabrielov’s rank Theorem.
We prove that Gabrielov’s rankTheorem admits three alternative formulations, which are of independent interest:
Theorem 1.7 (Variations of Gabrielov’s rank Theorem) . The following statementshold true: (I)
Let ϕ : A −→ B be a C -analytic morphism, where B is an integral domain. r( ϕ ) = r F ( ϕ ) = ⇒ r F ( ϕ ) = r A ( ϕ ) . (II) Let ϕ : A −→ B be a strongly injective morphism of analytic C -algebraswhere B is an integral domain. If f ∈ B is integral over b A then f isintegral over A . (III) Let f ∈ C { x, t } , where t is a single indeterminate. Assume that there isa non-zero g ∈ C J x, t K , such that f g ∈ C J x K [ t ] . Then there is a non-zero h ∈ C { x, t } such that f h ∈ C { x } [ t ] . A. BELOTTO DA SILVA, O. CURMI, AND G. ROND (IV)
Let f ∈ C { x, z } where z is a single indeterminate and n > . Set A := C J x K and B := C J x, z K ( x − x z ) . If the image of f in B is integral over A , then f is integral over C { x } . The proof of the above result is given in §§ I ) = ⇒ ( II ) = ⇒ ( III ) = ⇒ ( IV ) = ⇒ ( I ). The Theorem then immediatelyfollows because ( I ) is Gabrielov’s rank Theorem 1.4.One striking feature of Theorem 1.7 is that statements ( II, III, IV ) are intrin-sically algebraic. This contrasts with the statement of Gabrielov’s rank Theorem,which depends on the generic rank (a geometrical condition). It seems importantto clarify this relationship. We believe that the following open problem would behelpful in this investigation:
Problem 1.8.
Consider a family of local rings ( A n ) n ∈ N , where A n is a subring of K J x , . . . , x n K . It is natural to ask:(1) Under which hypothesis over ( A n ) n ∈ N does Gabrielov’s rank Theorem hold?(2) Under which hypothesis over ( A n ) n ∈ N are all the four statements in Theo-rem 1.7 equivalent?Note that the problem is also well-posed when K is a field of positive character-istic (see [Ro09] for the generalization of the geometric rank to fields of positivecharacteristic). Furthermore, if we consider a morphism ϕ : A −→ B where A and B are quotients of convergent power series rings by ideals generated by algebraicpower series, and if the components of ϕ are algebraic power series, then we alwayshave r( ϕ ) = r F ( ϕ ) = r A ( ϕ ) (see [Ro09, Theorem 6.7] for the general case, and[To76, Be77, Mi78] for partial cases).We finish this paragraph by pointing out that the statement of Theorem 1.7( II )above can be refined in the following way: Corollary 1.9.
Let ϕ : A −→ B be a morphism of analytic C -algebras where B isan integral domain. Let us assume that ϕ is strongly injective. If f ∈ B is algebraicover b A then f is algebraic over A . The proof of this result is given in §§ Connection with Zariski Main Theorem.
We now turn our attention to ZariskiMain Theorem, a classical result in algebraic geometry:
Zariski’s main Theorem ([Za48, Za50]) . Let A be a reduced local ring that isessentially finitely generated over a field k . Let A denote the integral closure of A in Frac( A ) (that is, the integral closure with respect to A −→ Frac( A ) ). Then theintegral closure of b A in Frac( b A ) coincides with the completion of A . Note that Theorem 1.7(II) and Corollary 1.9 can be seen as generalizations ofthe above result, where we replace the morphism A −→ Frac( A ) with a stronglyinjective morphism ϕ : A −→ B .1.2.4. Connection with elimination theory and completion.
In commutative algebraand in algebraic geometry, elimination theory is the study of elimination of vari-ables between polynomials. This is the main step in the resolution of polynomialequations. For example, in the case of linear equations, elimination theory reduces
ABRIELOV’S RANK THEOREM 7 to Gaussian elimination. In general, the main tools in elimination theory are theresultant and the Gröbner basis. Note that, unfortunately, there is no analogueof the resultant for power series, and the analogue of Gröbner basis, the standardbasis, is not as powerful for the objectives of elimination theory.The general situation is the following: Let x and y be two vectors of indetermi-nates and I an ideal of C { x, y } . The problem is to determine I ∩ C { x } . Note that,unlike in the polynomial case, we may have I ∩ C { x } = (0) even if ht( I ) is largerthan the number of indeterminates y i [Os16]. By Remark 1.17 below, we may evenhave I ∩ C { x } = (0) while I C J x, y K ∩ C J x K = (0). Therefore, an interesting questionis to determine under which hypothesis I C J x, y K ∩ C J x K is generated by I ∩ C { x } .This question has been investigated for the first time in [CPR19] where it isrelated to several other properties.In this context, given f ∈ C { x, y } where y is a single variable, it is important tounderstand under which conditions we may assume that f ∈ C { x } [ y ], up to multi-plication by a convergent unit. Such a result would allow us to adapt argumentsfrom elimination theory to the more general context of convergent power series.From this perspective, Theorem 1.7(III) provides a formal characterization of theabove condition.1.2.5. Connection with the Weierstrass preparation Theorem.
The Weierstrass prepa-ration Theorem is a very powerful tool in algebraic and analytic geometry. In thissubsection, we show how Gabrielov’s rank Theorem can also be seen as an extensionof the Weierstrass preparation Theorem for rings of convergent power series. Recallthat the usual form of the Weierstrass Theorem is the following one:
Theorem (Weierstrass preparation Theorem: usual formulation) . Let f be a for-mal (resp. convergent) power series in the indeterminates x , . . . , x n over C .Assume that f is x n -regular of order d , that is, f (0 , . . . , , x n ) = x dn × unit( x n ) .Then there exist unique formal (resp. convergent) power series a , . . . , a d in theindeterminate x ′ := ( x , · · · , x n − ) such that f ( x ) = (cid:0) x dn + a ( x ′ ) x d − n + · · · + a d ( x ′ ) (cid:1) × unit( x ) . Another classical form of the Weierstrass preparation Theorem is the followingone (see [Mal67] for instance):
Weierstrass preparation Theorem.
Let A −→ B be a morphism of analytic(resp. complete) C -algebras. Let m be the maximal ideal of A . Then B is finiteover A if and only if B/ m B is finite over A/ m = C . As a direct Corollary, we obtain the following case of Theorem 1.4:
Corollary 1.10 (Gabrielov’s rank Theorem for finite morphisms) . Let ϕ : A −→ B be an injective and finite morphism of analytic C -algebras where B is an integraldomain. Then b ϕ : b A −→ b B is injective and finite.Proof. Let m (resp. b m ) be the maximal ideal of A (resp. b A ). We have m b A = b m .Thus, if ϕ : A −→ B is finite, then A/ m −→ B/ m B is finite by the Weierstrasspreparation Theorem. But A/ m = b A/ b m and B/ m B = b B/ b m B . Hence b ϕ : b A −→ b B is again finite, by the Weierstrass preparation Theorem applied to b ϕ .Now, since ϕ is finite, we have dim( A ) = dim( B ) by [Mat89, Theorems 9.3.ii,9.4.ii]. Hence dim( b A ) = dim( A ) = dim( B ) = dim( b B ). But, since b A −→ b B is finite, A. BELOTTO DA SILVA, O. CURMI, AND G. ROND the induced morphism b A/ Ker( b ϕ ) −→ b B is also finite, thus dim( b A/ Ker( b ϕ )) =dim( b B ). Therefore dim( b A/ Ker( b ϕ )) = dim( b A ) and Ker( b ϕ ) is a height 0 prime ideal.But, since ϕ is injective and B is a domain, A is a domain, and b A is also a domain(c.f. [Ro18, Proposition 4.1] for example). This proves that Ker( b ϕ ) = (0) and b ϕ isinjective. (cid:3) Remark 1.11 (On the connection with Problem 1.8) . We claim that the Weier-strass preparation Theorem is a necessary condition for Gabrielov’s rank Theoremto hold in a family of real or complex rings ( A n ) n ∈ N , as asked in Problem 1.8. In-deed, let ϕ : A −→ B be an injective morphism between rings that are quotientsof rings A n , and assume that A/ m −→ B/ m B is finite. By the Weierstrass prepa-ration Theorem for complete local algebras, we have that b ϕ : b A −→ b B is finite.In particular any element f ∈ B is integral over b A . Therefore, if Theorem 1.7(II) is satisfied for the family ( A n ) n , we necessarily have that f is integral over A .Therefore if B = A n /I for some n and some ideal I of A n , and A n is a subring of K J x , . . . , x n K as in Problem 1.8, we have that the x i are integral over A , therefore B is integral over A .1.2.6. Convergent power series with support in strongly convex cones.
In generalroots of polynomials with coefficients in C J x K can be represented as Laurent Puiseuxseries with support in the translation of a rational strongly convex cone by a Theo-rem of MacDonald [McD95]. We will reformulate Gabrielov’s rank Theorem in thissetting. Before we need to give some definitions. Definition 1.12.
Let σ be a strongly convex rational cone containing R > n . Thismeans that σ has the form σ = { u ∈ R n | ∃ λ , . . . , λ k ∈ R > , u = λ v + · · · + λ k v k } where v , . . . , v k are given vectors with integer coordinates and σ does not containany non-trivial linear subspace of R n .For such a cone we denote by C J σ K the set of formal power series with supportin σ ∩ Z n , that is: C J σ K := ( f = X α ∈ σ ∩ Z n f α x α | f α ∈ C ) . More generally, if d ∈ N ∗ , we denote by C J σ ∩ d Z n K the set of formal power serieswith support in σ ∩ d Z n . Let us mention the following result: Theorem 1.13 (MacDonald Theorem [McD95]) . Let P ( z ) ∈ C J x K [ z ] be a monicpolynomial. Then there exists a strongly convex rational cone σ containing R n > and a positive integer d such that the roots of P ( z ) are in C J σ ∩ d Z n K . Since σ is a convex rational cone, there exists indeterminates u , . . . , u s anda binomial ideal I of C [ u ] such that C J σ K ≃ C J u K /I . Therefore we define theanalogue of the ring of convergent power series C { σ } as the subring of C J σ K whichis isomorphic to C { u } /I . This is also: C { σ } = ( f = X α ∈ σ ∩ Z n f α x α | ∃ C > , | f α ] < C | α | ∀ α ) . ABRIELOV’S RANK THEOREM 9
Theorem 1.7(II) has the following corollary about the Galois group of a polynomialwith formal power series coefficients:
Theorem 1.14.
Let P ( z ) ∈ C J x K [ z ] be a monic irreducible polynomial such thatthe roots of P ( z ) are in C J σ ∩ d Z n K , where σ is a strongly convex rational conecontaining R n > and d is a positive integer. If one of the roots of P is in C { σ ∩ d Z n } then the coefficients of P ( z ) are in C { x } . The proof of this result is given in §§ Examples.
In this section, we recall the classical examples of Osgood [Os16]and Gabrielov [Ga71].
Example 1.15 (Osgood’s example [Os16]) . Osgood showed the existence of amorphism ϕ : C { x , x , x } −→ C { u, v } such that(2) r( ϕ ) = 2 , r F ( ϕ ) = 3 , r A ( ϕ ) = 3 . Indeed, consider the following morphism: ϕ ( x ) = u, ϕ ( x ) = uv, ϕ ( x ) = uve v . We denote by b ϕ : C J x , x , x K −→ C J u, v K the morphism induced by ϕ . Given f ∈ Ker( b ϕ ), let us write f = P d ∈ N f d ( x ) where the f d ( x ) are homogeneous polynomialsof degree d (when they are not zero), so that:0 = b ϕ ( f ) = X d ∈ N f d ( u, uv, uve v ) = X d ∈ N u d f (1 , v, ve v )Therefore f d (1 , v, e v ) = 0 for every d , hence f d = 0 for every d since v and ve v arealgebraically independent over C . It follows that r F ( ϕ ) = r A ( ϕ ) = 3, while we caneasily check that r( ϕ ) = 2. In particular the map ϕ a : ( C , −→ ( C ,
0) definedby ϕ a ( u, v ) = ( u, uv, uve v ) sends a neighborhood of the origin onto a subset Z of C that is generically a complex manifold of dimension 2, but whose analytic orformal Zariski closure is C . Example 1.16 (Gabrielov’s example [Ga71]) . Gabrielov extended Osgood’s exam-ple, and provided a morphism ψ : C { x , x , x , x } −→ C { u, v } such that(3) r( ψ ) = 2 , r F ( ψ ) = 3 , r A ( ψ ) = 4 . which is build up from the observation that Osgood’s example ϕ is not well-behavedin terms of elimination theory, that is:(4) ϕ ( C { x } ) ( b ϕ ( C J x K ) ∩ C { u } , Indeed, we follow the heuristic that, even if x − x e x /x is not a power series, itsimage under ϕ should be 0. Let us consider a polynomial truncation of its formalpower series: f n := x − x n X i =0 i ! x i x i ! x n ∈ C [ x , x , x ] , ∀ n ∈ N . and note that ϕ ( f n ) = u n +1 v + ∞ X i = n +1 v i i ! , ∀ n ∈ N . It follows that ( n + 1)! ϕ ( f n ) is a convergent power series whose coefficients havemodule less than 1. Moreover when the coefficient of u k v l in the expansion of ϕ ( f n ) is nonzero, we have k = n + 1. This means that the supports of ϕ ( f n ) and ϕ ( f m )are disjoint whenever n = m . Therefore the power series h := X n ∈ N ( n + 1)! ϕ ( f n )is convergent since each of its coefficients has module less than 1. But b ϕ beinginjective, the unique element whose image is h is necessarily: b g : = X n ∈ N ( n + 1)! f n = X n ∈ N ( n + 1)! · x n ! x + b f ( x , x ) , Now, b g is a divergent power series and b ϕ ( b g ( x )) = h ( u, v ) ∈ C { u, v } . This showsthat (4) holds.Finally, consider the morphism ψ : C { x , x , x , x } −→ C { u, v } defined by ψ ( x ) = u, ψ ( x ) = uv, ψ ( x ) = uve v , ψ ( x ) = h ( u, v ) . By the above considerations, we see that x − b g ( x ) belongs to the kernel of b ψ . Infact one can show that Ker( b ψ ) = ( x − b g ( x )), while Ker( ψ ) = (0). Remark 1.17.
Note that Gabrielov’s example illustrates a case where the comple-tion operation does not commute with the elimination of indeterminates. Indeed,since Ker( b ψ ) = (0), there exist b k , . . . , b k ∈ C J x, u, v K such that( x − u ) b k + ( x − uv ) b k + ( x − uve v ) b k + ( x − h ( u, v )) b k ∈ C J x K r { } . This means that I C J x, u, v K ∩ C J x K = (0) where I denotes the ideal of C { x, u, v } generated by x − u, x − uv, x − uve v , x − h ( u, v ) . On the other hand, since Ker( ψ ) = (0), we see in a similar way that I ∩ C { x } = (0),as claimed. Remark 1.18 (Pathological real-analytic examples) . Variations of Osgood exam-ple have been used to provide the following list of pathological examples: • In [Pa89], Pawłucki provides an example of a subanalytic set (given by a nonregular morphism) which is neither formally nor analytically semi-coherent.In particular, this contradicted a result previously announced by Hironaka[H86]. • In [BP18], Bierstone and Parusiński show the existence of a proper real-analytic (non regular) mapping which can not be transformed into a map-ping with locally equidimensional fibers by global blowing ups (contrastingwith the complex case where the result holds true, as proved by Hironaka[H75]). • In [BdSB19], the first author and Bierstone show the existence of a properreal-analytic (non-regular) mapping which can not be monomialized viaglobal blowing ups in the source and target.2.
Ranks and transformations
General properties.
We follow the notations introduced in §§ Proposition 2.1 (Basic properties) . Let ϕ : A −→ B be a morphism of reduced C -analytic algebras. ABRIELOV’S RANK THEOREM 11 (1)
We have r( ϕ ) r F ( ϕ ) r A ( ϕ ) . (2) If r A ( ϕ ) = dim( A ) (resp. r F ( ϕ ) = dim( A ) ), ϕ is injective (resp. b ϕ isinjective). (3) Assume that B is an integral domain. Then r F ( ϕ ) = r A ( ϕ ) ⇐⇒ Ker( b ϕ ) = Ker( ϕ ) b A. Proof.
A rigorous proof of (1) is given in Lemma (1.5) [Iz89].Now assume that r A ( ϕ ) = dim( A ) and A is reduced. This means that Ker( ϕ )is an ideal of height 0. Since A has no non trivial nilpotents, Ker( ϕ ) = (0). Thesame proof works in the same way when r F ( ϕ ) = dim( A ). Indeed, by Artin approx-imation Theorem, b A is reduced when A is (see e.g. [Ro18, Proposition 4.1]), anddim( b A ) = dim( A ) (see [Mat89, Theorem 13.9] for example). This proves (2).For (3), let us remark that Ker( ϕ ) b A ⊂ Ker( b ϕ ). If B is an integral domain, b B is also an integral domain by Artin approximation Theorem, therefore Ker( ϕ ) andKer( b ϕ ) are prime ideals. By Artin approximation Theorem, Ker( ϕ ) b A is a primeideal of b A of same height as Ker( ϕ ). Therefore we haveKer( b ϕ ) = Ker( ϕ ) b A ⇐⇒ ht(Ker( b ϕ )) = ht(Ker( ϕ )) . This proves (3). (cid:3)
It is straightforward that the three ranks are invariant under isomorphisms. Theyare also invariant under some more general transformations, as shown in the follow-ing proposition:
Proposition 2.2.
Let ϕ : A −→ B be a morphism of reduced C -analytic algebrascorresponding to a morphism of germs of analytic sets Φ : ( Y, b ) −→ ( X, a ) . (1) Assume that B is an integral domain. Let σ : B −→ B be such that r( σ ) = dim( B ) , and B is an integral domain. Then all of the ranks of ϕ and σ ◦ ϕ coincide, that is, r( ϕ ) = r( σ ◦ ϕ ) , r F ( ϕ ) = r F ( σ ◦ ϕ ) and r A ( ϕ ) = r A ( σ ◦ ϕ ) . (2) Let τ : A −→ A be an injective finite morphism where A is an integraldomain, and assume that B is an integral domain. Then all of the ranks of ϕ and ϕ ◦ τ coincide.Proof. For (1), by Proposition 2.1 (1) we have that σ and b σ are injective, because B is an integral domain. Therefore Ker( b σ ◦ b ϕ ) = Ker( b ϕ ) and Ker( σ ◦ ϕ ) = Ker( ϕ ),and r F ( σ ◦ ϕ ) = r F ( ϕ ) and r A ( σ ◦ ϕ ) = r A ( ϕ ).Let us denote by ( Z, c ) the germ of analytic set associated to B . Since r( σ ) =dim( B ), the map σ a is an analytic diffeomorphism at a generic point in a neighbor-hood of c . It follows that r( ϕ ) = r( σ ◦ ϕ ).Finally, for (2), assume that τ is an injective finite morphism where A is anintegral domain. We have Ker( ϕ ◦ τ ) = Ker( ϕ ) ∩ A . Since B is an integral domain,Ker( ϕ ) and Ker( ϕ ◦ τ ) are prime ideals. Thus, by the Going-Down theorem for inte-gral extensions [Mat89, Theorem 9.4ii], we have that ht(Ker( ϕ ◦ τ )) ht(Ker( ϕ )),thus r A ( ϕ ) r A ( ϕ ◦ τ ). On the other hand, we have the equality r A ( ϕ ) = r A ( ϕ ◦ τ )because ht(Ker( ϕ ◦ τ )) = ht(Ker( ϕ )) by [Mat89, Theorem 9.3ii]. Now, since τ isfinite and injective, b τ is also finite and injective by Corollary 1.10. Moreover, wehave dim( b A ) − ht(Ker b τ ) = dim( b A ) = dim( A ) = dim( A ) = dim( b A ) since finite morphisms preserve the dimension and τ is injective. But ht(Ker( b τ )) = 0if and only if Ker( b τ ) = (0) because A is an integral domain. Thus, b τ is injectiveand r F ( ϕ ◦ τ ) = r F ( ϕ ).Eventually, if we denote by ( Z, c ) the germ of analytic set defined by A , we have τ a : ( X, a ) −→ ( Z, c ) is a finite map. Therefore r( ϕ ◦ τ ) = r( ϕ ). (cid:3) We now use the above Proposition to prove the following Lemma, which impliesthat Theorem 1.4 follows from Theorem 1.2:
Lemma 2.3.
Let ψ : A −→ B be a morphism of analytic C -algebras, where B is an integral domain. There exists an injective analytic morphism of C -algebras ϕ : C { x } −→ C { u } , where x = ( x , . . . , x m ) and u = ( u , . . . , u n ) , such that r( ψ ) = r( ϕ ) , r F ( ψ ) = r F ( ϕ ) and r A ( ψ ) = r A ( ϕ ) .Proof. Note that we can replace ψ by the morphism A Ker( ϕ ) −→ B induced by ψ , since the quotient by the Kernel clearly preserve all of the three ranks. Thuswe may assume that ψ is injective. By resolution of singularities there exists aninjective morphism of analytic C -algebras σ : B −→ B ′ which is a compositionof quadratic transformations and analytic isomorphisms such that B ′ = C { u } isregular. Next, by the Normalization Theorem for convergent power series (see[Na62, Theorem 45.5] or [dJPf00, Corollary 3.3.19]), there exists an injective finitemorphism τ : C { x } −→ A . We now set ϕ := σ ◦ ψ ◦ τ and we conclude by Proposition2.2. (cid:3) Monomial maps.
Thanks to Lemma 2.3, we can now focus on the regularcase, that is, when A = C { x } and B = C { u } . Apart form the isomorphisms, thetypical morphisms between smooth spaces that we use are those of the followingform:i) Power substitutions: C { u , . . . , u m } −→ C { e u , . . . , e u m } u e u a · · · · · · · · · u m e u a m m where a i ∈ N ∗ .ii) Quadratic transformations: C { u , . . . , u m } −→ C { e u , . . . , e u m } u e u e u u e u · · · · · · · · · u m e u m Let ϕ : A −→ B be a morphism of C -algebras, where A = C { x } and B = C { u } .It follows from Proposition 2.2 that: composition with a power substitution ora quadratic transformation in the target σ : B −→ B preserves all ranks; andcomposition with power substitutions in the source τ : A −→ A preserves allranks. Unfortunately, quadratic transformations in the source may not preservethe ranks: Remark 2.4 (On quadratic transformations in the source) . Let us consider themorphism ϕ : C { x, y, z } −→ C { u, v } defined by ϕ ( x ) = u , ϕ ( y ) = v and ϕ ( z ) = ABRIELOV’S RANK THEOREM 13 uve v , and a quadratic transformation τ : C { x , y , z } −→ C { x, y, z } defined by τ ( x ) = x , τ ( y ) = xy and τ ( z ) = z . Then we have ϕ ◦ τ ( x ) = u, ϕ ◦ τ ( y ) = uv, ϕ ◦ τ ( z ) = uve v . which is Osgood’s map (see Example 1.15). Thus we have r F ( ϕ ◦ τ ) = r A ( ϕ ◦ τ ) = 3while r F ( ϕ ) = r A ( ϕ ) = 2 (because Ker( ϕ ) and Ker( b ϕ ) are generated by z − xye y )and r( ϕ ) = r( ϕ ◦ τ ) = 2.Power substitutions and quadratic transformations are monomial morphisms.One basic but important property of these morphisms is the following one: Lemma 2.5.
Consider an n × n square matrix M = ( µ ij ) of natural numbers suchthat det( M ) = 0 , and the monomial map τ : C J x K −→ C J u K defined by: τ ( x i ) = u µ i = u µ i · · · u µ in n , i = 1 , . . . , n. If f ∈ C J x K is such that τ ( f ) ∈ C { u } , then f ∈ C { x } .Proof. Consider the formal expansions f = X α ∈ N n f α x α and τ ( f ) = X α ∈ N n f α u M · α = X β ∈ N n g β u β . By hypothesis, there exists two constants
A, B > | g β | AB | β | for every β ∈ N n . Let µ ∞ = k M k ∞ = max µ k,j . Since det( M ) = 0, we conclude that | f α | = | g Mα | AB | Mα | A ( B n µ ∞ ) | α | for every α ∈ N n , proving that f is convergent. (cid:3) The following result shows how we can use power substitutions and quadratictransformations in order to transform a given morphism of convergent power seriesrings into a morphism with a simpler form, but without changing the ranks:
Lemma 2.6 (Preparation of ϕ ) . Let ϕ : C { x , . . . , x n } −→ C { u , . . . , u n } be amorphism of convergent power series rings. There is a commutative diagram C { x } ϕ / / τ (cid:15) (cid:15) C { u } σ (cid:15) (cid:15) C { x } ϕ ′ / / C { u } where i) σ is a composition of quadratic transformations, power substitutions andisomorphisms; ii) τ is a composition of power substitutions and isomorphisms; iii) If r( ϕ ) = 1 , then ϕ ′ ( x ) = u and ϕ ′ ( x j ) = 0 for j = 2 , . . . , n . iv) If r( ϕ ) = n = 2 , then ϕ ′ ( x ) = u and ϕ ′ ( x ) = u a u b with a > and b ∈ Z > . v) If r( ϕ ) > and ϕ is injective, then (5) ϕ ′ ( x ) = u , ϕ ′ ( x j ) = u a j g j ( u ) , j = 2 , . . . , n where a j ∈ Z > , g j (0) = 0 and g j (0 , u , . . . , u n ) = 0 for j = 2 , . . . , n .In these conditions, we have r( ϕ ′ ) = r( ϕ ) , r F ( ϕ ′ ) = r F ( ϕ ) and r A ( ϕ ′ ) = r A ( ϕ ) . In particular, condition ( iii ) combined with Lemma 2.3 immediately implies thefollowing very particular case of Gabrielov’s rank Theorem:
Corollary 2.7 (Generic rank 1) . Let ϕ : A −→ B be a morphism of reducedanalytic C -algebras. If r( ϕ ) = 1 , then r F ( ϕ ) = r A ( ϕ ) = 1 .Proof of Lemma 2.6. We will prove the lemma by transforming step by step themorphism ϕ into the morphism ϕ ′ .Up to a linear change of coordinates in C { u } we may assume that the initialform of ϕ ( x ) evaluated at ( u , , . . . ,
0) is equal to Cu e for some e > C ∈ C ∗ .Let σ : C { u } −→ C { u } be the quadratic transform defined by σ ( u ) = u and σ ( u j ) = u u j for j = 2 , . . . , n . Then σ ◦ ϕ ( x ) = u e U ( u ) where U ( u ) ∈ C { u } is aunit. Let us replace ϕ by σ ◦ ϕ . Up to replacing x by U (0) x we may assume that U (0) = 1. Now, let V ( u ) ∈ C { u } be a convergent power series whose e -th power isequal to U ( u ). Let τ : C { x } −→ C { x } be the finite morphism (power substitution)given by τ ( x ) = x e , τ ( x j ) = x j for j = 2 , . . . , n . Replacing ϕ by ϕ ◦ τ , we mayassume that ϕ ( x ) = u V ( u ) where V ( u ) is a unit. Moreover by composing ϕ withthe inverse of the isomorphism of C { u } sending u onto u V ( u ), we may assumethat ϕ ( x ) = u .Let ϕ ( x j ) = ϕ j ( u ) ∈ C { u } the image of x j under ϕ and consider the analyticisomorphism: x x , x x j − ϕ j ( x , , j = 2 , . . . , n If r( ϕ ) = 1, we conclude that ϕ j ( u ) ≡ ϕ ) > ϕ is injective. We easily conclude that ϕ j ( u ) / ∈ C { u } for all j = 2 , . . . , n . Furthermore, because of the change of variables, we know that ϕ j ( u , , . . . ,
0) = 0, which implies that ϕ j ( u ) = u a j g j ( u ) for some a j > g j (0) = 0and g j (0 , u , . . . , u n ) = 0, proving (v).Finally, assume that n = 2. After composing ϕ with k quadratic transforma-tions of the form ( u , u ) ( u , u u ), for a sufficiently large k , we can supposethat g ( u ) = u b W ( u ), where b > W (0) = 0. After composing ϕ with theisomorphism whose inverse is defined by u u and u u W ( u ) /b , we havethe desired result.The last statement follows from Proposition 2.2. (cid:3) Gabrielov’s rank Theorem
Low dimensional Gabrielov’s rank Theorem.
Somehow surprisingly, themost difficult case in the proof of Gabrielov’s rank Theorem is the following:
Theorem 3.1 (Low-dimension Gabrielov I) . Let ϕ : C { x , x , x } −→ C { u , u } be an C -analytic morphism of convergence power series. Then r( ϕ ) = r F ( ϕ ) = 2 = ⇒ r A ( ϕ ) = 2 . Indeed, we deduce the Theorem 1.4 from Theorem 3.1 in the next subsection,following the same strategy as the one originally used by Gabrielov. Later, sections4 and 5 will be entirely dedicated to proving Theorem 3.1, where we deviate fromGabrielov’s original approach. This last part will involve a geometric setting andthe use of transcendental tools.There exists a particular case of Theorem 3.1 which admits a simple algebraicproof, namely when the generator of Ker( b ϕ ) is a quasi-ordinary polynomial. This ABRIELOV’S RANK THEOREM 15 particular case turns out to be crucial later on in the proof of Theorem 3.1. Wefinish this section by proving this result:
Proposition 3.2 (Quasi-ordinary case) . Let P ∈ C J x , x K [ y ] be a reduced monicnon-constant polynomial. Assume the following: i) ∆ P = x a x a × unit( x ) for some formal unit unit( x ) , ii) there exists a morphism ϕ : C { x, y } −→ C { u , u } with r( ϕ ) = 2 such that P ∈ Ker( b ϕ ) .Then P admits a non trivial monic divisor in C { x , x } [ y ] .Proof. The proof combines the Abhyankar-Jung Theorem, recalled in § ϕ ) = 2 and r F ( ϕ ) = 2 since P ∈ Ker( b ϕ ). Becauser F ( ϕ ) = 2, Ker( b ϕ ) is a height one prime ideal, thus a principal ideal, by [Mat89,Theorem 20.1]. Therefore any generator of Ker( b ϕ ) divides P . If r A ( ϕ ) = 2, thenKer( ϕ ) is height one prime ideal of C { x, y } , and Ker( ϕ ) C J x, y K = Ker( b ϕ ). Thusany generator f ( x, y ) of Ker( ϕ ) divides P . Since P is a monic polynomial, we musthave f (0 , y ) = 0. Therefore, by the Weierstrass Preparation Theorem, f ( x, y ) =unit × e P ( x, y ) where e P ( x, y ) ∈ C { x } [ y ] is a monic polynomial. Thus e P divides P ,and the proposition is proven in this case.Let us now prove that r A ( ϕ ) = 2. We denote by ϕ ( u ), ϕ ( u ) and ϕ ( u ) therespective images of x , x and y under ϕ . By Abhyankar-Jung Theorem, we canexpand P ( y ) as P = d Y i =1 (cid:16) y − b ξ i ( x /e , x /e ) (cid:17) . Furthermore, by Lemma 2.6 (iv) we may suppose that ϕ ( u ) = u and ϕ ( u ) = u a u b with b >
0. Therefore we may extend ϕ as a morphism ϕ ′ from C { x /e , x /e , y } to C { u /e , u /e } by defining ϕ ′ ( x /e ) := u /e , ϕ ′ ( x /e ) := u a/e u b/e . By Proposition 2.2 (2) we have r A ( ϕ ′ ) = r A ( ϕ ) . Therefore, if one of the b ξ i isconvergent, Ker( ϕ ′ ) = (0), thus r A ( ϕ ) = r A ( ϕ ′ ) = 2.By the above reduction, we may assume that e = 1 by replacing ϕ by themorphism: x u x u a u b y ϕ ( u e , u e )and P by P ( x e , x e , y ). Replacing in the original equation, we have, since P ∈ Ker( b ϕ ), d Y i =1 (cid:16) ϕ ( y ) − b ξ i ( u , u a u b ) (cid:17) = 0 . Hence, there is an index i such that b τ ( b ξ i )( u , u a u b ) ∈ C { u } . Thus, by Lemma 2.5, b ξ i ( x ) ∈ C { x } and r A ( ϕ ) = 2, as we wanted to prove. (cid:3) Reduction of Theorem 1.4 to Theorem 3.1.
The proof of Theorem 1.4 isdone by contradiction, following closely the ideas of Gabrielov [Ga73, Theorem 4.8].We note that we do not use in this section, at any point, a quadratic transformation τ : A −→ A in the source, c.f. Remark 2.4. We assume:( ∗ ) There exists a morphism ϕ : A −→ B of analytic C -algebras, where B is anintegral domain, such that r( ϕ ) = r F ( ϕ ) but r F ( ϕ ) < r A ( ϕ ). Suppose that ( ∗ ) holds true. Then, there exists an injectivemorphism ϕ : C { x } −→ C { u } , where x = ( x , . . . , x m ) and u = ( u , . . . , u n ), suchthat r( ϕ ) = r F ( ϕ ) = m − >
1, r A ( ϕ ) = m and Ker( b ϕ ) is a principal (nonzero)ideal.Indeed, by Lemma 2.3, there exists an injective morphism ϕ : C { x } −→ C { u } ,where x = ( x , . . . , x m ) and u = ( u , . . . , u n ), such that r( ϕ ) = r F ( ϕ ) >
1, butr F ( ϕ ) < r A ( ϕ ). Since r F ( ϕ ) < r A ( ϕ ) = m , we know that Ker( b ϕ ) = (0). Now,suppose that Ker( b ϕ ) is not principal or, equivalently, that its height is at least 2.By the Normalization Theorem for formal power series, after a linear change ofcoordinates, the canonical morphism π : C J x , . . . , x r( ϕ ) K −→ C J x K Ker( b ϕ )is finite and injective. Therefore the ideal p := Ker( b ϕ ) ∩ C J x , . . . , x r( ϕ )+1 K is anonzero height one prime ideal. Since C J x , . . . , x r( ϕ )+1 K is a unique factorizationdomain, p is a principal ideal (see [Mat89, Theorem 20.1] for example).Now, denote by ϕ ′ the restriction of ϕ to C { x , . . . , x r( ϕ )+1 } . By definitionKer( b ϕ ′ ) = p , thus r F ( ϕ ′ ) = r( ϕ ) + 1 − ϕ ) = r F ( ϕ ). Since ϕ is injective, ϕ ′ isinjective and r A ( ϕ ′ ) = r( ϕ ) + 1. Moreover, since π is finite, by Proposition 2.2 wehave: r( ϕ ′ ) = r( b ϕ ′ ) = r( b ϕ ) = r( ϕ ) . Therefore we replace ϕ by ϕ ′ and we assume that r( ϕ ) = r F ( ϕ ) = m − A ( ϕ ) = m , as we wanted to prove. Suppose that ( ∗ ) holds true. Then, we claim that there existsan injective morphism ϕ : C { x } −→ C { u } , where x = ( x , . . . , x n +1 ) and u =( u , . . . , u n ) (that is, m = n + 1), such that r( ϕ ) = r F ( ϕ ) = n and Ker( b ϕ ) is aprincipal (nonzero) ideal (in particular, r F ( ϕ ) = n < n + 1 = r A ( ϕ )).We consider the morphism given in the 1st Reduction. Up to a linear changeof coordinates in u , we can suppose that the rank of the Jacobian matrix of ϕ evaluated in ( u , . . . , u r , , . . . ,
0) is still equal to r := r( ϕ ). Let us denote by ϕ thecomposition of ϕ with the quotient map C { u } −→ C { u , . . . , u r } , which satisfiesr( ϕ ) = r( ϕ ). Now, by Proposition 2.1 we have:1 = m − r( ϕ ) > m − r F ( ϕ ) = ht(Ker( b ϕ )) > ht(Ker( b ϕ )) = 1 . The last inequality comes from the fact that Ker( b ϕ ) ⊂ Ker( b ϕ ). This shows thatht(Ker( b ϕ )) = 1. Therefore Ker( b ϕ ) = Ker( b ϕ ) since both are prime ideals of heightone. We note, furthermore, that ϕ is injective. Indeed, if Ker( ϕ ) = (0), take anonzero f ∈ Ker( ϕ ), and note that f ∈ Ker( b ϕ ) by the previous equality, whichimplies that f ∈ Ker( ϕ ), a contradiction. We can assume that n = r and m = n + 1,as was required. ABRIELOV’S RANK THEOREM 17
Suppose that ( ∗ ) holds true. Then, we claim that there existsan injective morphism ϕ : C { x } −→ C { u } , where x = ( x , x , x ) and u = ( u , u )(that is, m = 3 and n = 2), such that r( ϕ ) = r F ( ϕ ) = 2 and Ker( b ϕ ) is a principal(nonzero) ideal.(Note that the 3rd Reduction contradicts Theorem 3.1, providing the desiredcontradiction, and proving Theorem 1.4.)We consider the morphism given in the 2nd Reduction. Let P be a generator ofKer( b ϕ ). After a linear change of coordinates we may assume that P is a Weierstrasspolynomial with respect to x n +1 =: y and that ϕ ( x i ) is not constant for i =1 , . . . , n . Our goal is to reduce the dimension n by restriction to hyperplanes in thecoordinates x . More precisely, we rely in two results about generic sections: Theorem 3.3 (Abhyankar-Moh Reduction Theorem, [AM70, pp. 31]) . Let F ∈ C J x K be a divergent power series, and denote by Λ the subset of C of all constants λ ∈ C such that F ( λx , x , . . . , x n ) ∈ C { x , . . . , x n } is convergent. Then Λ hasmeasure zero. Theorem 3.4 (Formal Bertini Theorem, [Ch58]) . Let k be an uncountable field, n > and x = ( x , . . . , x n ) . Let P ( y ) ∈ k J x K [ y ] be irreducible. Then there isan at most countable subset A ⊂ k , such that P ( cx + dx , x , . . . , x n , y ) remainsirreducible in k J x , . . . , x n K [ y ] for every c ∈ k r A and every d ∈ k transcendentalover k P,c where k P,c denotes the field extension of the prime field of k generated bythe coefficients of P ( x, y ) and c (in particular k P,c is a countable field).
We will say that a hyperplane x − P ni =2 λ i x i = 0 is generic if λ ∈ C n − can bechosen outside a subset of measure zero. Note that both theorems demand a generichyperplane section in the variable x . Our task now is to modify the morphism(without changing its ranks) in order to guarantee that it can be restricted to ageneric hyperplane section.Indeed, by Lemma 2.6, we may assume that r( ϕ ) > ϕ ( x j ) with j = 1, . . . , n satisfy the normal form given in Lemma 2.6 (v). The image of ϕ ,nevertheless, does not yet necessarily include generic hyperplanes in x . In orderto deal with this issue we use a trick of Gabrielov (whose idea we illustrate in theconcrete example 3.5 below).Let us perform the trick. Up to composing ϕ with a quadratic transformation u u and u j u u j for j = 2 , . . . , n , we may suppose that a j ∈ Z > in the normal forms (5) of Lemma 2.6 (v). Furthermore, up to making powersubstitutions of the form x j x α j j with α j = Q k = j a k for every j = 2 , . . . , n , wecan suppose that a j = a > j . Finally, we consider the powersubstitution x x a +11 . All these operations preserve the ranks of the morphismby Proposition 2.2. We therefore have obtained the following normal form: ϕ ( x ) = u a +11 , ϕ ( x j ) = u a g j ( u ) , j = 2 , . . . , n where g j (0) = 0 and g j (0 , u , . . . , u n ) = 0. Now, let us consider a linear function h λ ( x ) = x − P nj =2 λ j x j with λ j ∈ C for j = 2 , . . . , n . Note that: ϕ x − n X i =2 λ i x i ! = u a u − n X i =2 λ j g j ( u ) ! =: u a g λ ( u ) . We claim that for a generic choice of λ ∈ C n − , the hypersurface V λ := ( g λ ( u ) = 0)is not contained in the set of critical points of the morphism ϕ a : ( C n , −→ ( C n +1 ,
0) (where we recall that ϕ a ∗ = ϕ ), which we denote by W .Indeed, on the one hand, outside of a proper analytic subset Γ ⊂ C n − , weknow that g λ (0) = 0, g λ (0 , u , . . . , u n ) = 0 and ∂ u g λ (0) = 0. Therefore, bythe implicit function Theorem, the equation g λ ( u ) = 0 admits a nonzero solution u = ξ λ ( u , . . . , u n ).On the other hand, assume by contradiction that V λ ⊂ W for λ ∈ Λ ⊂ C n − ,where Λ is of positive measure. Since W is a proper analytic subset of C n , W contains only finitely many hypersurfaces. Therefore, when λ runs over Λ, V λ runsover finitely many hypersurfaces. We define an equivalence relation on Λ by λ ≃ e λ if V λ = V e λ . Then Λ is the disjoint union of the (finitely many) equivalence classes.Therefore, at least one of these equivalence classes is not included in an affinehyperplane. We denote it by Λ . Then(6) n X j =2 ( e λ j − λ j ) · g j ( u ) | V λ = (cid:16) g λ ( u ) − g e λ ( u ) (cid:17) | V λ ≡ , ∀ λ, e λ ∈ Λ . Since Λ is not included in an affine hyperplane, we may choose λ , e λ (1) , . . . , e λ ( n ) such that the vectors λ − e λ (1) , . . . , λ − e λ ( n ) are linearly independent. Therefore, (6)applied to the λ − e λ ( k ) implies that g j ( u ) | V λ ≡
0, so that u | V λ ≡
0. In other words,( g λ ( u ) = 0) ⊂ ( u = 0), so that ξ λ ( u , . . . , u n ) = 0, which is a contradiction.Therefore, for a generic choice of λ ∈ C n − , the induced morphism: ψ λ : C J x , . . . , x n K [ y ]( x − P ni =2 λ i x i ) −→ C J u , . . . , u n K ( u − P ni =2 λ j g j ( u ))is such that r( ψ λ ) = n −
1. Finally, let us assume that n >
3. By Theorem 3.4 and3.3, the polynomial P remains irreducible and divergent in C J x , . . . , x n K [ y ] / ( x − P ni =2 λ i x i ) for a generic choice of λ = ( λ , . . . , λ n ) ∈ C n − . We conclude, therefore,that r F ( ψ λ ) = n − A ( ψ λ ) = n . By repeating this process, we obtain thedesired morphism with n = 2. Example 3.5 (On Gabrielov’s trick) . (1) Consider the morphism ϕ : C { x , x , x , y } −→ C { u , u , u } given by ϕ ( x ) = u , ϕ ( x ) = u u , ϕ ( x ) = u u and ϕ ( y ) = f ( u ) . Consider a hyperplane H λ = ( x − λ x − λ x ) where ( λ , λ ) = (0 , ϕ ( x − λ x − λ x ) = u U ( u ) where U (0) = 0. It followsthat the restriction of ϕ to H λ induces a morphism: ψ λ : C { x , x , x , y } ( x − λ x − λ x ) −→ C { u , u , u } ( u ) = C { u , u } which is constant equal to zero, so that r( ψ λ ) = 0.(2) (Gabrielov’s trick). In order to solve the above issue, we perform a powersubstitution in the source (which preserves all ranks by Proposition 2.2).More precisely, we consider x = x , so that we now have ϕ ( x ) = u . Itnow follows that: ϕ ( x − λ x − λ x ) = u ( u − λ u − λ u ) ABRIELOV’S RANK THEOREM 19 and the restriction of ϕ to H λ induces a morphism: ψ λ : C { x , x , x , y } ( x − λ x − λ x ) −→ C { u , u , u } ( u − λ u − λ u )and we can easily verify that r( ψ λ ) = 2 = r( ϕ ) − Proof of Formal Bertini Theorem.
Before giving the proof of the theoremwe make the following remarks:
Remark 3.6 (On the Formal Bertini Theorem) . (1) A stronger version of the above result was originally stated in [Ch58] (where A is assumed to be finite), but we were not able to verify the proof. Wefollow the same strategy as Chow to prove the above result.(2) Tougeron proposes an alternative proof of the Formal Bertini Theorem in[To72, page 349] via a “Lefschetz type" Theorem. The proof is geometricand does not adapt in a trivial way to the formal case.(3) (Counterexample of formal Bertini for n = 2). Consider the irreduciblepolynomial P ( x, y ) = y − ( x + x ). For every λ ∈ C , we have that: P ( x, λx, y ) = y − (1 + λ ) x = ( y − x p λ )( y + x p λ )is a reducible polynomial. Therefore, there is no Formal Bertini Theoremfor n = 2.For c ∈ k we set R ( c ) := k J x K [ z ]( zx − ( x + cx )) = k J x K (cid:20) x + cx x (cid:21) . The ideal generated by the x i is a prime ideal of R ( c ), denoted by p ( c ), and thecompletion of R ( c ) p ( c ) is k ( z ) J x , . . . , x n K .We begin by giving the proof of the following Proposition, given as a lemma in[Ch58]: Proposition 3.7.
Let k be an uncountable field and n > . Let P ( y ) ∈ k J x K [ y ] bean irreducible monic polynomial. Then there is a countable subset A ⊂ k such that,for all c ∈ k r A , P ( y ) is irreducible in b R ( c ) p ( c ) [ y ] .Proof. More generally, for c , . . . , c s ∈ k , s distinct elements with s >
2, we set R ( c , . . . , c s ) := k J x K [ z , . . . , z s ] / ( z x − ( x + c x ) , . . . , z s x − ( x + c s x )) . The ideal generated by the x i is a prime ideal of R ( c , . . . , c s ), denoted by p , and thecompletion of R ( c , . . . , c s ) p is isomorphic to k ( z , . . . , z s ) J x , . . . , x s K , where the z i satisfy the following relations over k (for every i , j , k , ℓ with i = j and k = ℓ ):(7) c i z j − c j z i c i − c j = c k z ℓ − c ℓ z k c k − c ℓ ; z i − z j c i − c j = z ℓ − z k c ℓ − c k . Therefore the completion of R ( c , . . . , c s ) p is isomorphic to k ( z , z ) J x , . . . , x s K ,and z and z are algebraically independent over k (we may replace z and z byany other z i , z j ). Let us fix s = 2. We have, in b R ( c , c ) p : z x = x + c x , z x = x + c x , x = c z − c z c − c x , x = z − z c − c x . Therefore, for an element f ∈ b R ( c , c ) p , we have f ∈ R if and only if(8) f = X α ∈ N n f α z α z α x α · · · x α n n , where [ α > α or α > α = ⇒ f α = 0]Let P ( y ) ∈ k J x K [ y ] be irreducible and assume that there exist an uncountableset I and distinct elements c i ∈ k , i ∈ I , such that P ( y ) is reducible in b R ( c i ) p ( c i ) [ y ].We fix two such c i that we denote by c and c . By the previous discussion, wemay assume that the b R ( c i ) p ( c i ) are all embedded in k ( z , z ) J x , . . . , x n K . Sincethere is only finitely many ways of splitting a monic polynomial into the productof two monic polynomials, we may assume that we have the same factorization P ( y ) = P ( y ) P ( y ) in all the b R ( c i ) p ( c i ) [ y ].Let f be a coefficient of P ( y ) or P ( y ) in b R ( c , c ) p ( c i ) , that we write f = X β ∈ N n − f β x β · · · x β n n where the f β ∈ k ( z , z ). Since f ∈ b R ( c ) p ( c ) and b R ( c ) p ( c ) ≃ k ( z ) J x , . . . , x n K ≃ k ( z ) J z x , x , . . . , x n K we have that f β ∈ k ( z )[ z ] for all β , and deg z ( f β ) β for all β such that f β = 0.By symmetry, we have that f β ∈ k [ z , z ] and deg z ( f β ) β for all β such that f β = 0. Now let us choose one more c i , that we denote by c . By (7), we have z = c − c c − c z + c − c c − c z . By replacing z by c − c c − c z + c − c c − c z in the f β , we obtain the coefficients g β of theexpansion f = X β ∈ N n − g β x β · · · x β n n as an element of k ( z , z ) J x , . . . , x n K . In particular, we have that g β ∈ k [ z , z ] anddeg z i ( g β ) β for i = 1 or 3, for all β such that g β = 0. For a given β , we havedeg z ( g β ) deg z ,z ( f β )where deg z ,z denotes the total degree in z , z . This inequality may be strict,as some cancellations may occur. But, for a given β , there is a finite set A β ⊂ k (possibly empty) such that c − c c − c / ∈ A β = ⇒ deg z ( g β ) = deg z ,z ( f β ) . We remark that the map H : c ∈ k c − c c − c is injective. Therefore the set A := [ β ∈ N n − H − ( A β )is at most countable. Since I is uncountable, we may choose c / ∈ A . Therefore forsuch a c , we have ∀ β, deg z ( g β ) = deg z ,z ( f β ) . Now, by the previous discussion done for the f β , we also have deg z ( g β ) β .Therefore, for every β , we have deg z ,z ( f β ) β . Thus, by (8), we see that f ∈ R .This argument applies to any coefficient of P ( y ) or P ( y ). Hence, we have that ABRIELOV’S RANK THEOREM 21 P ( y ), P ( y ) ∈ R [ y ]. This contradicts the assumption that P ( y ) is irreducible in R [ y ]. (cid:3) Now we can give the proof of Theorem 3.4:
Proof of Theorem 3.4.
We apply Proposition 3.7 to P ( y ). Let c / ∈ A . By construc-tion, the image of P ( y ) in S ( c )[ y ] is P ( cx + zx , x , . . . , x n , y )where S ( c )[ y ] is identified with k ( z ) J x , . . . , x n K [ y ]. In fact, we have P ( cx + zx , x , . . . , x n , y ) ∈ k P,c ( z ) J x , . . . , x n K [ y ] . Therefore, if we replace z by any element of k that is transcendental over k P,c , wehave that P ( cx + zx , x , . . . , x n , y ) is irreducible. (cid:3) Proof of the low-dimensional Gabrielov Theorem
Geometrical framework.
In order to prove Theorem 3.1, we will use geo-metric arguments involving transcendental tools. In particular, the article changespace and we use essentially geometric language instead of algebraic. We start byfixing the notation.Given a point a ∈ C n , when n is clear form the context we write O a for the ring ofanalytic germs O ( C n ) a , and b O a for its completion. Given an analytic germ f ∈ O a ,we denote by b f a the Taylor series associated to f at the point a . In particular, theBorel mapping is the morphism of local rings: T a : O a −→ b O a f b f a A coordinate system centered at a is a collection of functions x = ( x , . . . , x n ) whichgenerate the maximal ideal m a of O a , in which case we recall that O a is isomorphicto C { x } . We define, similarly, coordinate systems b x = ( b x , . . . , b x n ) of b O a , and wenote that b O a is isomorphic to C J b x K . We say that a coordinate system b x of b O a is convergent if there exists a coordinate system x of O a such that b x i = T a ( x i ) for i = 1 , . . . , n . Whenever b x is a convergent coordinate system, we abuse notation andwe identify it with x .Let Φ : Y −→ X be a complex analytic map between two smooth analyticspaces. Given b ∈ Y and a ∈ X where a = Φ( b ), we will denote by Φ b theassociated analytic map germ Φ b : ( Y, b ) −→ ( X a ). We denote by Φ ∗ b : O a −→ O b the morphism of local rings defined by ∀ f ∈ O a , Φ ∗ b ( f ) := f ◦ Φ b . Then we denote by b Φ ∗ b : b O a −→ b O b the completion morphism of Φ ∗ b . Following thenotation introduced in the introduction, we note that (Φ ∗ b ) a is the localization ofthe morphism Φ to b , that is, (Φ ∗ b ) a : ( Y, b ) → ( X, a ).When ϕ : A −→ B is a morphism of local rings, and P ∈ A [ y ] is a polynomialwith coefficients in A , P = p + p y + · · · + p d y d , we will use the following abuse ofnotation: ϕ ( P ) = ϕ ( p ) + ϕ ( p ) y + · · · + ϕ ( p d ) y d ∈ B [ y ] . Concretely, we will use this notation with A = O a or b O a , and B = O b or b O b . Geometrical formulation of low-dimension results.
We now re-phraseTheorem 3.1 and Proposition 3.2 in the geometrical context. Instead of a morphism ϕ : C { x , x , y } −→ C { u , u } such that r( ϕ ) = 2 and r F ( ϕ ) = 2, we work witha morphism of germs Φ : ( C , a ) −→ ( C , Φ( a )) such that Φ ∗ = ϕ , r(Φ ∗ ) = 2 andr F (Φ ∗ ) = 2. In order to simplify the notation, we always assume that Φ( a ) ∈ a × C (which is always possible up to a translation in the target) and that there exists aformal polynomial P ∈ b O a [ y ] such that b Φ ∗ ( P ) = 0, that is, Ker( b Φ ∗ ) = ( P ) (whichwe can always suppose by Weierstrass preparation).We introduce the following definition: Definition 4.1 (Convergent factor) . Let P ∈ b O a [ y ] be a non-constant monic poly-nomial. We say that P admits a convergent factor (at a ) if there exists an analyticnon-constant monic polynomial q ∈ O a [ y ] such that T a ( q ) = b q a divides P .We are ready to reformulate Theorem 3.1: Theorem 4.2 (Low-dimension Gabrielov II) . Let P ∈ b O a [ y ] be a monic polynomial,where a ∈ C . Suppose that there exists an analytic morphism Φ : ( C , a ) −→ ( C , Φ( a )) , generically of rank , such that Φ( a ) ∈ { a } × C and b Φ ∗ a ( P ) = 0 . Then P admits a convergent factor. In particular, if P is a formally irreduciblepolynomial, then P is an analytic polynomial. (Note that Theorem 4.2 immediately implies Theorem 3.1.)It is easy to see that, in Theorem 3.1, we can always suppose that the polynomial P ∈ b O a [ y ] is reduced. In particular, the discriminant of P (with respect to theprojection ( x, y ) ( x )) is a nonzero formal power series which we denote by∆ P ∈ C J x K . In what follows, quasi-ordinary singularities (that is, when ∆ P ismonomial) will play and important role. We start by distinguishing two cases ofquasi-ordinary singularities: Definition 4.3 (Monomial discriminant) . Let P ∈ b O a [ y ] be a reduced monic non-constant polynomial. We say that the discriminant ∆ P is formally monomial ifthere exists a coordinate system x = ( x , x ) of b O a such that:∆ P = x α · W ( x ) = x α · x α · W ( x )where W ( x ) ∈ b O a is a unit. We say that the discriminant ∆ P is analyticallymonomial if x = ( x , x ) is a coordinate system of O a . Example 4.4 (Formally vs analytically monomial) . Suppose that:∆ P = x · x − ∞ X n =1 n ! · x n ! . On the one hand, there is no (formal) unit U ( x , x ) such that U ( x , x )( x − P ∞ n =1 n ! · x n ) is convergent. Indeed, by the unicity of the preparation given by theWeierstrass preparation Theorem, this would imply that U and x − P ∞ n =1 n ! · x n are convergent power series, which is not the case. Therefore ∆ P is not analyticallymonomial. On the other hand, after the formal change of coordinates: b x = x , b x = x − ∞ X n =1 n ! · x n . ABRIELOV’S RANK THEOREM 23 we conclude that ∆ P ( b x ) = b x · b x , which is formally monomial.We are ready to reformulate Proposition 3.2: Proposition 4.5 (Final case) . Let P ∈ b O a [ y ] be a reduced monic polynomial, where a ∈ C , whose discriminant ∆ P is analytically monomial. Suppose that there existsan analytic morphism Φ : ( C , a ) −→ ( C , Φ( a )) , generically of rank , such that Φ( a ) ∈ { a } × C and b Φ ∗ a ( P ) = 0 , then P admits a convergent factor. Note that Proposition 3.2 immediately implies Proposition 4.5.We remark that Proposition 4.5 is a particular case of Theorem 4.2. The proofof Theorem 4.2 consists in reducing to the quasi-ordinary case via blowing ups inthe target of Φ. As we have remarked in 2.4, a blowing up does not preserve theformal and analytic ranks of the analytic germ. We must make global argumentsover the blowed-up space, as we discuss in the next section.4.3.
Blowing ups and the inductive scheme.
We consider an analytic manifold N (which is eventually assumed of dimension 2) and a simple normal crossing divisor F over N . An admissible blowing up (for the couple ( N, F )) is a blowing up: σ : ( e N , e F ) −→ ( N, F )whose center C is connected and has normal crossings with F , that is, at everypoint a ∈ C , there exists a coordinate system x of O a and t ∈ { , . . . , n } such that C = ( x = · · · = x t = 0) and F is locally given as a finite union of hypersurfaces( x i = 0). In particular, note that if C is a point, then the blowing up is admissible.A sequence of admissible blowing ups is a finite sequence of morphisms:( C , a ) = ( N , a ) ( N , F ) σ o o · · · σ o o ( N r , F r ) σ r o o and we fix the convention that σ : ( N r , F r ) −→ ( N , F ) denotes the compositionof the sequence.The proof of Theorem 4.2 demands an argument in terms of the history of theexceptional divisors, so it is convenient to introduce notation to keep track of thehistory explicitly. More precisely, consider a sequence of admissible blowing ups( σ , . . . , σ r ). Note that, for every j ∈ { , . . . , r } , the exceptional divisor F j is asimple normal crossing divisor which can be decomposed as follows: F j = F (0) j ∪ F (1) j ∪ · · · ∪ F ( j ) j , ∀ j = 1 , . . . , r where F (0) j is the strict transform of F , and for every k ∈ { , . . . , j } , the divisor F ( k ) j is an irreducible and connected component of F j which is uniquely defined viathe following recursive convention: • If k = j , then F ( j ) j stands for the exceptional divisor introduced by σ j ; • If k < j , the divisor F ( k ) j is the strict transform of F ( k ) j − by σ j .The proof of Theorem 4.2 will follow from combining Proposition 4.5 with thefollowing result, as we show in §§ Proposition 4.6 (Inductive scheme) . Let a ∈ C and consider a non-constantreduced monic polynomial P ∈ b O a [ y ] . Consider a sequence of admissible blowingups ( C , a ) = ( N , a ) ( N , F ) σ o o · · · σ o o ( N r , F r ) σ r o o we set σ := σ ◦ · · · ◦ σ r , and we assume that: i) ∀ b ∈ σ − ( a ) , we have that b σ ∗ b (∆ P ) is formally monomial, ii) ∃ k ∈ { , . . . , r } , ∃ b ∈ F ( k ) r such that P b = b σ ∗ b ( P ) has a convergent factor q .Then P admits a convergent factor whose pullback under σ is equal to q in a neigh-borhood of b . Remark 4.7.
In Example 4.4, we have illustrated that the discriminant of P isnot analytically monomial in general. In fact, even after a sequence of blowing ups,∆ P is not analytically monomial in general. Indeed, let us choose the polynomial P given in Example 4.4. If we consider the quadratic transformation ( x , x ) =( z , z z ) we get b σ ∗ (∆ P ) = z ( z − − X n > n ! z n − . Therefore, b σ ∗ (∆ P ) is not analytically monomial at the point of coordinates ( z , z ) =(0 ,
1) as shown in Example 4.4. A straightforward induction shows that this is againthe case after finitely many blowing ups. Therefore we need to be careful when wereduce the proof of Theorem 4.2 to Propositions 4.5 and 4.6 (c.f. Case II in §§ §§ Proposition 4.8 (Semi-Global extension) . Let a ∈ C and consider a non-constantreduced monic polynomial P ∈ b O a [ y ] . Consider a sequence of admissible blowingups ( C , a ) = ( N , a ) ( N , F ) σ o o · · · σ o o ( N r , F r ) σ r o o we set σ := σ ◦ · · · ◦ σ r , and we assume that: i) ∀ b ∈ σ − ( a ) , b σ ∗ b (∆ P ) is formally monomial, ii) ∃ b ∈ F (1) r such that P b = b σ ∗ b ( P ) has a convergent factor q .Then, there exists an open neighbourhood U (1) r of F (1) r , and a convergent polynomial q ∈ O U (1) r [ y ] such that, at every point c ∈ F (1) r , the polynomial b q c divides P c := b σ ∗ c ( P ) and either P c = b q c or the quotient P c / b q c does not admit a convergent factor. In Corollary 5.18 below, we prove a more precise version of the above Proposition.Indeed, § P ∈ O a [ y ] is convergent, then it is clear that a convergent factor q at a point a , is also a convergent factor of P on a neighborhood of a . When P ∈ b O a [ y ] isdivergent, then this property still holds over “fibers", that is: Proposition 4.9 (Convergent factor along fibers) . Let a ∈ C n and Φ : M −→ C n be an analytic map, generically of maximal rank, where M is smooth. Let P ∈ b O a [ y ] be a non-constant monic polynomial and suppose that there exists b ∈ Φ − ( a ) ABRIELOV’S RANK THEOREM 25 such that P b = ϕ b ( P ) admits a convergent factor q at b . Then, there exists aneighbourhood U of b such that, for every point c ∈ U ∩ Φ − ( a ) , the polynomial q is a convergent factor of P c . We prove Proposition 4.9 in §§ Proof of Gabrielov’s low dimension Theorem (Reduction of Theorem4.2 to Proposition 4.6).
The discriminant ∆ P is a formal curve in ( C , a ), soit admits a resolution of singularities via blowing ups of points which are alwaysconvergent centres. In other words, there exists a sequence of (analytic) pointblowing ups: ( C , a ) = ( N , a ) ( N , F ) σ o o · · · σ o o ( N s , F s ) σ s o o such that for every b ∈ σ − ( a ), the pulled-back discriminant b σ ∗ b (∆ P ) is formallymonomial (but not necessarily analytically monomial). Note that any further se-quence of points blowing ups preserve this property, and we may compose thissequence with further blowing ups of points if necessary. Now, let( C , a ) × C = ( M , a × C ) ( M , E ) σ × Id o o · · · σ × Id o o ( M s , E s ) σ s × Id o o be the associated sequence of admissible blowing ups over C . We can now showthat there exists a sequence of point blowing ups in the source of the morphism Φ:( C , a ) = ( L , a ) ( L , D ) λ o o · · · λ o o ( L s , D s ) λ s o o where λ i denotes a finite sequence of blowing ups, including length zero (so, theidentity); and mappings Φ i : ( L i , D i ) −→ ( M i , E i ) for i = 1 , . . . , s , such that thefollowing diagram commutes: L (cid:15) (cid:15) L (cid:15) (cid:15) λ o o · · · λ o o L s − s − (cid:15) (cid:15) λ s − o o L s Φ s (cid:15) (cid:15) λ s o o M M σ × Id o o · · · σ × Id o o M s − σ s − × Id o o M sσ s × Id o o where Φ = Φ. Indeed, this result follows from usual resolution of indeterminacy ofmaps: let I be the reduced ideal sheaf whose support is the first center of blowingup C in M , and consider its pullback J = Φ ∗ ( I ). Let λ : L −→ L denote thesequence of point blowing ups that principalize J ; we conclude by the universalproperty of blowing ups the existence of the morphism Φ : L −→ M . It is enoughto repeat this argument for the entire sequence.Now, since Φ is generically of maximal rank and the λ i are sequences of pointblowing ups, we conclude that Φ s is generically of maximal rank. We denote by λ the composition of the λ i . Let c ∈ λ − ( a ) and denote by Φ s ( c ) = ( b , b ) its image.Note that: b Φ ∗ s c ( P b ) = b Φ ∗ s c ◦ b σ ∗ b ( P ) = b λ ∗ c ◦ b Φ ∗ a ( P )by hypothesis.We now consider the two following cases: Case I:
Suppose that ∆ P b is analytically monomial. In this case we do not needto make any other subsequent blowing ups (and r = s when we apply Proposition4.6). Indeed, all hypothesis of Proposition 4.5 are satisfied, so we conclude that P b b γ ∆ P B = 0 Π ◦ Φ s ( L s ) F s N s Π ◦ Φ r ( L r ) F r N r b γ ∆ P = 0( τ s +1 ◦ · · · ◦ τ r ) Figure 1.
Proof of Theorem 4.2: case II.admits a convergent factor. This implies that all hypothesis of Proposition 4.6 aresatisfied, so we conclude that P admits a convergent factor, as we wanted to prove. Case II:
Suppose that ∆ P b is not analytically monomial, but only formally mono-mial. In this case, we make further blowing ups in order to reduce to Case I (and r > s when we apply Proposition 4.6).Let Σ ⊂ L s be the analytic subset of L s where Φ s is not of maximal rank (notethat this set is the support of the ideal generated by the determinant of the two-by-two minors of the Jacobian of Φ s ). Since Σ is an analytic curve, there exists a curve˜ γ : ( C , −→ ( L s , c ) which intersects Σ ∪ Φ − ( E s ) only at c . We set Π : M s −→ N s the canonical projection. Then, taking γ = Π ◦ Φ s ◦ ˜ γ , we obtain an analytic curve on( N s , b ) which intersects the exceptional divisor F s only at b . It follows that γ and(∆ P b = 0) can not have flat contact (otherwise, (∆ P b = 0) would be convergent),so by a sequence of point blowing ups we can separate the strict transform of γ and (∆ P b = 0). We are now in Case I when we center at the point b ′ given by thestrict transform of γ . Indeed, at b ′ , the germ defined by ∆ P b ′ is equal to the germdefined by one irreducible component of the exceptional divisor.4.5. The induction Scheme (Reduction of Proposition 4.6 to Proposition4.8 and 4.9).
The proof of Proposition 4.6 follows by induction on the lexico-graphical order of ( r, k ). Note that the first step of the induction, that is when r = k = 0, is tautological. We now fix ( r, k ) and we assume that the Propositionis true whenever ( r ′ , k ′ ) < ( r, k ). We divide the proof in two parts, depending if k = 1 or k > Case I: k = 1 . By Proposition 4.8, there exists an open neighbourhood U (1) r of F (1) r ,and a convergent polynomial q ∈ O U (1) r [ y ] such that, at every point c ∈ F (1) r , thepolynomial b q c divides P c = b σ ∗ c ( P ) and, furthermore, either P c = b q c or the quotient P c / b q c does not admit a convergent factor. Note that, since F r is connected and U (1) r is open, by Proposition 4.9, at each connected component of F r r F (1) r , thereexists a point in this component, say c j , where b σ ∗ b ( P ) admits a convergent factor. ABRIELOV’S RANK THEOREM 27 F (1)1 W j V a j F (1) r F ( j ) r U (1) r c j c σ ◦ · · · ◦ σ r σ a σ Figure 2.
Proof of Proposition 4.6: case I.We now consider the geometrical picture after only the first blowing up σ , whichhas exceptional divisor F = F (1)1 . Let { a , . . . , a l } be all the points in F which arecentres of subsequent blowing ups. Note that σ ◦ · · · ◦ σ r : ( N r , F r ) −→ ( N , F )is an isomorphism at every point of F r { a , . . . , a l } . There exists, therefore, anopen neighbourhood V of F r { a , . . . , a l } , and an analytic polynomial e p ∈ O V [ y ]such that: σ ∗ r ◦ · · · ◦ σ ∗ ( e p ) = q Now, let us denote by P j := ( b σ ) ∗ a j ( P ) for j = 1 , . . . , l , which is a non-constantmonic polynomial in b O a j [ y ]. We consider the sequence of blowing ups σ (2) := σ ◦ · · · ◦ σ r . Note that the pull-back of the discriminant ∆ P j is everywhere formallymonomial, since it coincides with the pull-back of ∆ P by the entire sequence σ .Furthermore, since P admits a convergent factor at a point in every connected com-ponent of F r r F (1) r , we conclude that P j admits a convergent factor (at some point,say c j ) after its composition with σ (2) . It follows that P j satisfies all conditionsof Proposition 4.6 with ( r ′ , k ′ ) such that r ′ r −
1. By induction, P j admits aconvergent factor of maximal degree p j , defined in a neighborhood W j of a j . Fur-thermore, the degree of p j must be the same as the degree of e p , since they coincideafter pull-back by σ (2) at a point of F r r F (1) r , by the inductive assumption.Finally, since p j is convergent in a neighborhood W j , there exists a point b j ∈ W j ∩ F (1)1 where p j and e p are well-defined. Since these polynomials are convergent,have the same degree, and e p is the convergent factor of maximal degree of P ,we conclude that e p = p j at b j . It follows that e p extends in a neighbourhood U := V ∪ lj =1 W j of F , and it formally divides b σ ∗ ( P ) everywhere in F . Therefore,by Lemma 4.10 given below, e p = σ ∗ ( p ), where p ∈ O a [ y ] formally divides P , as wewanted to prove. Case II: k > . Let a k − denote the center of the blowing up σ k , and consider: P k − := \ ( σ ◦ · · · ◦ σ k − ) ∗ a k − ( P ) , and note that P k − is a non-constant monic polynomial which belongs to b O a k − [ y ].We consider the sequence of blowing ups σ ( k ) := σ k ◦ · · · ◦ σ r . Note that the pull-back of the discriminant ∆ P k − is everywhere formally monomial, since it coincideswith the pull-back of ∆ P by the entire sequence σ . Furthermore, since P admits F ( j ) k − F ( k ) r a k − b W k − σ k ◦ · · · ◦ σ r ca σ Figure 3.
Proof of Proposition 4.6: case II.a convergent factor over the exceptional divisor created by the blowing up σ k , weconclude that P k admits a convergent factor after its composition with σ ( k ) at somepoint c ∈ F ( k ) r . It follows that P k satisfies all hypothesis of Proposition 4.6 with( r ′ , k ′ ) such that r ′ r − k < r −
1. By induction, we conclude that P k − admitsa convergent factor p k − ∈ O a k − [ y ], defined in some neighborhood W k − of a k − .Therefore, by Proposition 4.9, there exists a point b ∈ F ( j ) k − , for some j k − b σ ∗ b ( P ) admits a convergent factor. We conclude by induction. Lemma 4.10.
Let σ : ( M, E ) −→ ( C , be the blowing up with center , andsuppose that there exists an analytic function h : U −→ C defined in an openneighbourhood of U of E . Then there exists an analytic germ f : ( C , −→ C suchthat h = f ◦ σ .Proof. Let a be a point in E . Up to making a linear change of coordinates, we cansuppose without loss of generality that a is the origin of the x -chart, that is: x = u x = u · v and a = (0 ,
0) in this coordinate system. Since h is analytic in a neighborhood U of E , h is constant along E by Liouville’s Theorem, that is, h ( u, v ) = h + uh ( u, v )with h ∈ C . By doing this at any point of E , we see that h can be extendedas an analytic function on U . By repeating this process we have that, locally at a , h = f ( u, uv ) for some formal power series f . By Lemma 2.5, the series f isconvergent. (cid:3) Convergence of factors along fibers (Proof of Proposition 4.9).
Theproof is divided in two steps, depending on the nature of Φ − ( a ): Step I:
Suppose that Φ − ( a ) = E is a SNC divisor. Let ( v, w ) = ( v, w , . . . , w n ) bea coordinate system centered at b such that ( v = 0) ⊂ E . We can write: x i = v α i Ψ i ( v, w ) , i = 1 , . . . , n where Ψ = (Ψ , . . . , Ψ n ) is an analytic morphism defined in some open neighbor-hood U of b , and Ψ i (0 , w ) = 0 for all i = 1 , . . . , n . Without loss of generality, wecan suppose that there exists a disc D ⊂ C such that U = D n and that the coef-ficients of the polynomial q are convergent over U . Next, we set α = ( α , . . . , α n ), ABRIELOV’S RANK THEOREM 29 and let A ∈ b O a be a fixed function. We consider the expansion of A in terms of α -homogeneous polynomials: A = ∞ X i =0 A i ( x ) where A i ( x α ) is an homogenous polynomial of degree i This implies that:Φ ∗ b ( A ) = ∞ X i =0 v i ψ ( A i ) = ∞ X i =0 v i ∞ X j =0 v j a ij ( w ) = ∞ X k =0 v k X i + j = k a ij ( w ) = ∞ X k =0 v k b k ( w )where, since A i are polynomials and Ψ is convergent in U = D n , we concludethat a ij ( w ) are analytic functions defined on D n − . Moreover, since for every k ,the function b k ( w ) is a finite sum of functions a ij ( w ), we conclude that b k ( w ) arealso analytic functions defined over D n − . Denoting by O D n − the ring of analyticfunctions defined over D n − , we conclude that b k ( w ) ∈ O D n − for every k ∈ N , sothat Φ ∗ b ( A ) ∈ O D n − J v K . Since the choice of A ∈ b O a was arbitrary, we concludethat P b ∈ O D n − J v K [ y ]. In particular, for every c ∈ ( v = 0) ∩ D n , we get that P c = P b as elements of O D n − J v K [ y ]. Furthermore, the factor q also belongs to O D n − J v K [ y ], and it follows from the Euclidean division that P b = q · Q + R , where R ∈ O D n − J v K [ y ] has an identically zero formal expansion at b . Since the ring O D n − is of convergent series, this implies that R ≡
0. It follows that q divide P c ,at every point c ∈ ( v = 0) ∩ U . We conclude the Proposition by remarking that E is a SNC divisor and the choice of the hypersurface ( v = 0) ⊂ E was arbitrary. Step II:
Let Σ := Φ − ( a ); since the morphism Φ is generically of maximal rank,we conclude that Σ is a proper analytic subvariety of M . Consider a resolutionof singularities of Σ, that is, an analytic morphism σ : ( M ′ , E ′ ) −→ ( M, Σ) ofmaximal rank such that σ − (Σ) = E ′ is a SNC divisor. From Step I, at every point b ′ ∈ σ − ( b ), there exists a neighbourhood U b ′ where σ b ′ ( q ) is a convergent factorof P b ′ . Since Σ is an analytic subvariety of M , there exists an open neighbourhood U of b where U ∩ Σ is connected. Since σ is proper, furthermore, up to shrinking U we can suppose that(9) σ − ( U ) ⊂ [ b ′ ∈ σ − ( b ) U b ′ . Now given a point c ∈ Σ ∩ U , suppose by contradiction that q is not a factor of P c , that is, the formal division P c = q c · Q + R has a nonzero remainder R ∈ b O c [ y ].It follows that, at every point c ′ ∈ σ − ( c ) we have that σ c ′ ( R ) := R c ′ = 0, whichimplies that q c ′ does not divide P c ′ . But c ′ ∈ E ∩ σ − ( U ), leading to a contradictionwith (9). It follows that q formally divides P c at every point c ∈ Σ ∩ U , as we wantedto prove. 5. Semi-Global extension of convergent factors
Semi-global extension overview (Proof of Proposition 4.8).
This sub-section contains the full strategy to prove Proposition 4.8. In order to motivateeach object, we leave the proofs and development of the necessary supporting tech-niques, namely Theorems 5.7, 5.15, 5.17 and Proposition 5.12, to subsections 5.2,5.3, 5.4 and 5.5.We start by providing the adequate algebraic setting for the discussion. Moreprecisely, following Tougeron [To90], we build up a subring of the algebraic closure of C ( x ) which captures geometrical properties necessary to address Proposition 4.8.Our presentation is at first general (that is, n ∈ N ), and we specialize it to the case n = 2 when it becomes convenient for the presentation.5.1.1. Preliminaries on valuation rings.
We consider the ring of power series C J x K where x = ( x , . . . , x n ) and we denote by C (( x )) its field of fractions. We denoteby ν the ( x )-adic valuation on C J x K . The valuation ν extends to C (( x )) by defining ν ( f /g ) = ν ( f ) − ν ( g ) for every f , g ∈ C J x K , g = 0. We consider the valuation ring V ν associated to it, that is V ν := { f /g | f, g ∈ C J x K , ν ( f ) > ν ( g ) } = { F ∈ C (( x )) | ν ( F ) > } . We denote by b V ν its completion. Classically elements of b V ν can be represented asformal series A = P k ∈ N A k where the A k are in V ν and, if A k = 0, ν ( A k ) = k .For B ∈ V ν we have B = f /g where f , g ∈ C J x K . We expand f = P k > k f k , g = P k > k g k where the f k and g k are homogeneous polynomials of degree k and f k , g k = 0. Therefore we have B = f k g k X k>k f k f k ! X k>k g k g k ! − Therefore the elements A ∈ b V ν are of the form: A = X k ∈ N a k ( x ) b k ( x ) , where a k and b k are homogeneous polynomials such that deg( a k ) − deg( b k ) = k . Definition 5.1 (Weighted-homogenous polynomial) . Let z , . . . , z r be indetermi-nates, and let ω , . . . , ω r ∈ Q > . We say that a polynomial Γ( x, z ) ∈ C [ x, z ] is( ω , . . . , ω r ) -weighted homogeneous if Γ( x, z ω , . . . , z ω r r ) is homogeneous. Definition 5.2 (Homogeneous elements) . A homogeneous element γ is an elementof an algebraic closure of C ( x ), satisfying a relation of the form Γ( x, γ ) = 0 forsome ω -weighted homogeneous polynomial Γ( x, z ), where ω ∈ Q > . Furthermore,if Γ( x, z ) is monic in z , we say that γ is an integral homogeneous element . In thiscase, ω is called the degree of γ .Given an integral homogeneous element γ of degree ω , there exists an extensionof the valuation ν , still denoted by ν , to the field C ( x )[ γ ], defined by ν d X k =0 a k ( x ) γ k ! = min { ν ( a k ) + kω } . where d is the degree of the field extension C ( x ) −→ C ( x )[ γ ]. In particular, theabove property justifies our use of the valuation ν (instead of naively using thenotion of order, which would not extend to integral homogeneous elements).More generally, given homogeneous elements γ = ( γ , . . . , γ r ) of degrees ω = ( ω ,. . . , ω r ), there exists an extension of the valuation ν , still denoted by ν , to the field C ( x )[ γ ], defined by ν X k ,...,k r a k ( x ) γ k = min { ν ( a k ) + k ω + · · · + k r ω r } ABRIELOV’S RANK THEOREM 31 where the indices k j run over { , . . . , d j } , where d j is the degree of the field extension C ( x )[ γ , . . . , γ j − ] −→ C ( x )[ γ , . . . , γ j ] . We denote by V ν,γ the valuation ring of ν defined on C (( x ))[ γ ]. We note that V ν,γ is a local ring, and we denote by b V ν,γ its completion. We remark that the imageof V ν,γ or of b V ν,γ under the valuation ν is the subgroup Γ ν,γ of Q generated by 1, ω , . . . , ω r . This group being a finitely generated Z -module, it is a discrete group,therefore V ν,γ and b V ν,γ are discrete valuation rings.Note that all elements of b V ν,γ are written as finite sums: X k ,...,k r a k ( x ) γ k , where a k ( x ) ∈ Frac( b V ν ) and ν ( a k ( x ) γ k ) > . Definition 5.3 (Initial term) . For a nonzero element ξ ∈ b V ν,γ , we can write ξ = X k ∈ Q > ∩ Γ ν,γ ξ k , where ξ k ∈ C (( x ))[ γ ] are ν -homogenous terms of degree k The initial term of ξ , denoted by in ν ( ξ ), is defined as ξ k where k = min { k ∈ Q > ∩ Γ ν,γ | ξ k = 0 } , which is well defined because Γ ν,γ is a finitely generated subgroup of Q .5.1.2. Projective rings and a Newton-Puiseux-Eisenstein Theorem.
We are nowready to define the rings of functions which we are interested in:
Definition 5.4 (Projective rings) . Let h be a homogeneous polynomial. We denoteby P h (( x )) the subring of b V ν characterized by the following property: A ∈ P h (( x )) ifthere exists k ∈ Z , α , β ∈ N and a k ( x ) homogeneous polynomials for k > k sothat: A = X k > k a k ( x ) h αk + β , where ν ( a k ) − ( αk + β ) ν ( h ) = k, ∀ k > k We denote by P h J x K the subring of P h (( x )) of elements A whose initial term k belongs to Z > . When γ is an integral homogeneous element, we denote by P h J x, γ K the subring of b V ν,γ , whose elements ξ are of the form: ξ = d X k =0 A k ( x ) γ k , where A k ∈ P h (( x )) and ν ( A k ( x ) γ k ) > , k = 0 , . . . , d. Remark 5.5.
Let us remark that the family ( P h J x K ) h is a directed set, since, fortwo homogeneous polynomials h and h , we have P h i J x K ⊂ P h h J x K for i = 1 , . Remark 5.6 (Geometrical properties of projective rings) . (1) (Invariance by linear coordinate changes). Let h be a homogenous polyno-mial, γ be an homogenous integral element and Φ : ( C n , −→ ( C n ,
0) bea linear coordinate change. Note that e h := h ◦ Φ is an homogenous poly-nomial and that P h J x K (respectively P h (( x )) and P h J x, γ K ) is isomorphic to P e h J x K (respectively P e h (( x )) and P e h J x, γ K ). (2) (Blowing ups) We specialize to the case n = 2. Let σ : ( N, F ) = ( N r , F r ) −→ ( C ,
0) be a sequence of point blowing ups and let b ∈ F (1) r be a point. Wecan find two different normal forms, depending on the nature of b :(a) Suppose that b does not belong to the intersection of two exceptionaldivisors. Then, up to a linear change of coordinates in x , there existsa system of local coordinates ( v, w ) centered at b such that: x = v, x = vw and, if A ∈ P h J x K , we obtain: A b := b σ ∗ b ( A ) := X k > a k (1 , w ) h (1 , w ) αk + β v k . so that b σ ∗ b : P h J x K −→ C ( w ) J v K is a well-defined morphism. In par-ticular, if b does not belong to the strict transform of ( h = 0) (so h (1 , = 0) then A b ∈ b O b and b σ ∗ b : P h J x K [ y ] −→ b O b is well-defined.(b) Suppose that b belongs to the intersection of two components of theexceptional divisor. Then, up to a linear change of coordinates in x ,there exists local coordinate system ( v, w ) centered at b and c ∈ Z > such that: x = vw c , x = vw c +1 , and, if A ∈ P h J x K , we obtain: A b := b σ ∗ b ( A ) := X k > a k (1 , w ) h (1 , w ) αk + β ( w c v ) k . Our interest in these rings is justified by the following version of Newton-Puiseux-Eisenstein Theorem, proved in subsection 5.2:
Theorem 5.7 (Newton-Puiseux-Eisenstein) . Let P ( x, y ) ∈ C J x K [ y ] be a monicpolynomial. There exists an integral homogeneous element γ , and a homogeneouspolynomial h ( x ) , such that P ( x, y ) factors as a product of degree monic polyno-mials in y with coefficients in P h J x, γ K . The following result is an easy, but convenient, reformulation of Theorem 5.7:
Corollary 5.8 (Newton-Puiseux-Eisenstein factorization) . Let P ∈ C J x K [ y ] be amonic polynomial. Then, there is a homogenous polynomial h and integral homoge-nous elements γ i,j , such that P can be written as (10) P ( x, y ) = s Y i =1 Q i , and Q i = r i Y j =1 ( y − ξ i ( x, γ i,j )) where (i) the Q i ∈ P h J x K [ y ] are irreducible in b V ν [ y ] , (ii) for every i , there are A i,k ( x ) ∈ P h (( x )) , for k k i such that ξ i ( x, γ i,j ) = k i X k =0 A i,k ( x ) γ ki,j (iii) for every i , the γ i,j are distinct conjugates of an homogeneous element γ i ,that is, roots of its minimal polynomial Γ i over C ( x ) . ABRIELOV’S RANK THEOREM 33
Proof.
The first equality is a direct consequence of Theorem 5.7. Fix Q an irre-ducible factor of P in P h J x K [ y ] and let Frac( P h J x K ) ֒ → K be a normal field exten-sion containing all the roots of Q . Because Q is irreducible, for every j , there is τ j ∈ Aut ( K / Frac( P h J x K )) such that τ j ( ξ ) = ξ j . Now, seeing ξ as an element ofFrac( P h J x K )[ γ ], this gives τ j ( ξ ( x, γ )) = ξ ( x, τ j ( γ )). But since C ( x ) ⊂ Frac( P h J x K ), τ j ( γ ) is also a root of the minimal polynomial of γ over C ( x ). (cid:3) Projective Convergent rings.
In order to prove Proposition 4.8, we will showthat if P admits a convergent factor after a sequence of blowing ups, then a certainnumber of the polynomials Q i in equation (10) are “convergent”. We start bymaking the latter notion precise, that is, we introduce a subring P h { x } of P h J x K which formalizes the notion of convergence in P h J x K . More precisely: Definition 5.9 (Projective convergent rings) . Let h be a homogeneous polynomial.We denote by P h { x } , the subring of P h J x K characterized by elements A ∈ P h J x K such that(11) X k > a k ( x ) ∈ C { x } , where A = X k > a k ( x ) h αk + β . Remark 5.10.
We recall that a power series f = P α ∈ N n f α x α , is convergent, ifand only if there exists A, B > ∀ α ∈ N n , | f α | A · B | α | . Moreover, if we expand f as f := X k ∈ N f k ( x )where f k ( x ) are homogeneous polynomials of degree k , then f ∈ C { x } if, and onlyif, there exists a compact neighbourhood K of the origin and constants A, B > z ∈ K | f k ( z ) | AB | k | , ∀ k ∈ N . Definition 5.11.
For f analytic on some compact set K ⊂ C n , we set k f k K := sup z ∈ K | f ( z ) | . Note that it is not clear that P h { x } is well-defined, since the characterizationof its elements seems to depend on the choice of the representation of A ∈ P h J x K in power series, which is not unique. The following Proposition, whose proof wepostpone to subsection 5.3, addresses this point and shows that P h { x } is well-defined: Proposition 5.12 (Independence of the representative) . Let h , h be homoge-neous polynomials and consider elements A ∈ P h J x K and A ∈ P h J x K such that A = A when they are considered as elements of b V ν . Then A ∈ P h { x } if andonly if A ∈ P h { x } . In particular: P h { x } ∩ C J x K = C { x } . The algebraic definition of P h { x } captures a crucial geometrical property forthis work, namely the “generic” convergence of elements of A ∈ P h { x } after a pointblowing up. More precisely: Lemma 5.13 (Geometrical characterization of P h { x } ) . Let A ∈ P h J x K and con-sider a sequence of point blowing ups σ : ( N, F ) −→ ( C n , . Let b be a point of F (1) r which is not on the strict transform of ( h = 0) , nor on any other componentof F . Then A ∈ P h { x } , if and only if, A b := b σ ∗ b ( A ) is a convergent power series.Proof. We start by noting that the definition of P h { x } is invariant by linear changesof coordinates in x , c.f. Remark 5.6(1). Therefore, since b is only on F (1) r , we canuse the normal form of Remark 5.6(2.a) for σ b . Assume that A ∈ P h { x } as in (11).The degree of a k ( x ) is linear in k , say ak + b with a, b ∈ N . Since P a k ( x ) ∈ C { x } ,there exists a compact neighbourhood K of b and a constant C > k a k (1 , w ) k K C k . Since b is not on the strict transform of ( h = 0), h (1 , w ) =0, and, up to shrinking K , we can suppose that inf ( v,w ) ∈ K k h (1 , w ) k = h > (cid:13)(cid:13)(cid:13)(cid:13) a k (1 , w ) h (1 , w ) αk + β (cid:13)(cid:13)(cid:13)(cid:13) K h β (cid:18) Ch α (cid:19) k and we easily conclude that A b is a convergent power series.Next, suppose that A b is convergent and let us prove that the formal power series G := P k > k a k ( x ) ∈ C J x K is convergent. Indeed, we note that: B ( z, w ) := X k > a k (1 , w ) z k = h β (1 , w ) · A b ( h α (1 , w ) · z, w ) ∈ C { z, w } . Now, by definition, the degree of the polynomials a k ( x ) is an affine function in k , say ak + b where a , b ∈ N . It therefore follows that b σ ∗ b ( G ) = z b · B ( z a , w ) isa convergent power series. It now follows from Lemma 2.5 that G is convergent,finishing the proof. (cid:3) Formal extensions.
We now turn our attention to the study of the behaviourafter blowing up of A ∈ P h { x } at a point b in the strict transform of ( h = 0). Westart by studying it formally: Definition 5.14 (Formal extension) . Let A ∈ P h J x K and consider a sequence ofpoint blowing ups σ : ( N, F ) −→ ( C n , b ∈ F (1) r , we say that A extends formally at b if the composition A b := b σ ∗ b ( A ) belongs to b O b . Moreover, wesay that A extends analytically at b if A b belongs to O b .Given A ∈ P h J x K , we know by Lemma 5.13 that A extends to every point b ∈ F (1) r which does not belong to the strict transform of ( h = 0) or to the intersection oftwo divisors. But, under the hypothesis of Proposition 4.8, the results hold trueover every point b ∈ F (1) r , that is: Theorem 5.15 (Semi-global formal extension) . Let P ( x, y ) ∈ C J x K [ y ] be a monicreduced polynomial, and let P = Q si =1 Q i be the factorization of P as a product ofirreducible monic polynomials of P h J x K [ y ] given by Theorem 5.7. Suppose that n = 2 and let σ : ( N, F ) −→ ( C , be a sequence of point blowing ups so that, at everypoint b ∈ F (1) r , the pulled-back discriminant b σ ∗ b (∆ P ) is formally monomial. Then,for every point b ∈ F (1) r , the polynomials Q i extend formally at b to a polynomialwhich we denote by b σ ∗ b ( Q i ) . Furthermore, the extension is compatible with thefactorization of P , that is Q ri =1 b σ ∗ b ( Q i ) = b σ ∗ b ( P ) . ABRIELOV’S RANK THEOREM 35
The formal extension property can be combined with analytic continuation typearguments in order to obtain:
Lemma 5.16 (Analytic continuation of formal extensions) . Let A ∈ P h { x } and σ : ( N, F ) −→ ( C , be a sequence of point blowing ups. Suppose that A formallyextends to b ∈ F (1) r . Then A extends analytically at b .Proof. The result easily follows from Lemma 5.13 whenever b is not in the stricttransform of ( h = 0) or in the intersection of more than one exceptional divisor.In the general case, using the normal forms of Remark 5.6(2), we can suppose thatthere exists a coordinate system ( v, w ) centered at b and c ∈ Z > such that: x = v · w c , x = v · w c +1 it follows from the hypothesis that there exists polynomials b k ( w ) for k > k suchthat A b = X k > k v k w ck a k (1 , w ) h (1 , w ) αk + β = X k > k v k · b k ( w ) =: B ( v, w )that is, b k · h (1 , w ) αk + β = w ck · a k (1 , w ). We claim that B is a convergent powerseries. Indeed, let us denote by h (1 , w ) = w d u ( w ), where u (0) = 0, and considera closed ball B ( b , r ) of radius 1 > r > w ∈ B ( b ,r ) | u ( w ) | > C for somepositive C .For every polynomial b ( w ), by the maximum principle, we have k b ( w ) k B ( b ,r ) = | b ( z ) | = 1 r d | b ( z ) z d | r d k b ( w ) w d k B ( b ,r ) for some z ∈ C , | z | = r .Therefore, for every polynomial b ( w ), we get that: k b k B ( b ,r ) k b · u k B ( b ,r ) k u − k B ( b ,r ) r d k b · h (1 , w ) k B ( b ,r ) k u − k B ( b ,r ) Cr d k b · h (1 , w ) k B ( b ,r ) Now, by Remark 5.10, there is a constant
D > k a k (1 , w ) k B ( b ,r ) D k for every k . Combining both inequalities, we get: k b k k B ( b ,r ) C β · ( C α D ) k , ∀ k > B is a convergent power series by Remark 5.10. (cid:3) Local-to-Semi-global convergence of factors.
As a consequence of the previousdiscussion, if P ∈ C J x K [ y ] is a monic polynomial and Q ∈ P h { x } [ y ] is a factor of P then, under the conditions of Proposition 4.8, Q b := b σ ∗ b ( Q ) is a convergent factorof P b := b σ ∗ b ( P ) at every point b ∈ F (1) r . It remains to show that there exists sucha factor Q . This is the subject of the next result: Theorem 5.17 (Local-to-Semi-global convergence of factors) . Let P ∈ C J x K [ y ] bea monic reduced polynomial, and h be a homogeneous polynomial as in Theorem5.7. Suppose that n = 2 and let σ : ( N, F ) −→ ( C , be a sequence of pointblowing ups. Suppose that at every point b ∈ F (1) r , the pulled-back discriminant b σ ∗ b (∆ P ) is formally monomial. Suppose that there exists a point b ∈ F (1) r such that P b := b σ ∗ b ( P ) admits a convergent factor. Then P admits a non-constant factor Q ∈ P h { x } [ y ] such that either P/Q is constant, or b σ ∗ b ( P/Q ) admits no convergentfactor at every point b ∈ F (1) r . Proof of Proposition 4.8.
We have now collected all necessary ingredients inorder to prove the following result, which immediately yields Proposition 4.8:
Corollary 5.18 (Semi-global extension of convergent factors) . Let P ( x, y ) ∈ C J x K [ y ] be a monic reduced polynomial, and let P = Q si =1 Q i be the factorization of P asa product of irreducible monic polynomials of P h J x K [ y ] given in Theorem 5.7. Let σ : ( N, F ) −→ ( C , be a sequence of point blowing ups, and suppose that: (i) At every point b ∈ F (1) r , the pulled-back discriminant b σ ∗ b (∆ P ) is formallymonomial. (ii) There exists b ∈ F (1) r where P b := b σ ∗ b ( P ) admits a convergent factor.Then, up to re-indexing the polynomials Q j , there exists an index t > , a neigh-bourhood U (1) r of F (1) r and analytic polynomials e q j ∈ O U (1) r [ y ] for j = 1 , . . . , t suchthat, for every point b ∈ F (1) r , we have that T b ( e q j ) = b σ ∗ b ( Q j ) . Finally, denote by e q = Q ti =1 e q i . Then, at every point b ∈ F (1) r , the quotient polyno-mial P b /T b ( e q ) is either constant, or does not admits a convergent factor.Proof. Consider the factorization (10) given by Corollary 5.8: P ( x, y ) = s Y i =1 Q i ( x, y ) , where Q i ( x, y ) ∈ P h J x K [ y ] , i = 1 , . . . , s From Theorem 5.15, we know that Q i extends formally to every point b of F (1) r forevery i = 1 , . . . , s . It follows from Theorem 5.17 that there exists t s such that Q i ∈ P h { x } [ y ] for every 1 i t . Now, by Lemma 5.16, Q i extends analyticallyat every point b ∈ F (1) r for every i t . This implies that there exists a polynomial e q i defined in a neighbourhood U (1) r of F (1) r which formally coincides with b σ ∗ b ( Q i ) atevery point b ∈ F (1) r . Finally, it is immediate from the above construction that t isconstant along F (1) r , which implies that P/ Q ti =1 Q i admits no further convergentfactors, finishing the proof. (cid:3) Newton-Puiseux-Eisenstein Theorem.
The proof of Theorem 5.7 is donevia an induction argument in terms of the degree of P . Note that the case ofdeg( P ) = 1 is trivial (with h = 1 and γ = 1), while degree( P ) = 2 still admits anelementary proof: Remark 5.19 (Elementary proof when deg( P ) = 2) . If deg( P ) = 2, then we canwrite P ( x, y ) = P ( x ) + P ( x ) y + y and obtain an explicit formula for their roots.More precisely, the discriminant ∆ P ∈ C J x K , and can be written as:∆ P := X k > k δ k ( x )where δ k = in(∆ P ) and δ k ( x ) are homogeneous elements for every k > k . Itfollows that the roots of P ( x, y ) = 0 are given by: ξ ± = − P ± √ ∆ P − P ± p δ k ( x )4 · vuut X k>k δ k ( x ) δ k ( x ) ABRIELOV’S RANK THEOREM 37 and we can easily verify that ξ ± ∈ P h J x, γ K , where h := in(∆ P ) = δ k ( x ) and γ := p in(∆ P ) = p δ k ( x ).In general, it is not possible to choose h = in(∆ P ) as claimed (but not proven)in [To90].As we will see, a proof by induction on the degree of P demands manipulationsof several integral elements at the same time. We start by a couple of useful resultsabout homogeneous elements: Lemma 5.20 (Degree compatibility under extensions) . Let P ( x, z , . . . , z k +1 ) ∈ C [ x, z , . . . , z k +1 ] be a ( ω , . . . , ω k +1 ) -weighted homogeneous polynomial. Let γ i behomogeneous elements of degree ω i , for i k , such that deg z k +1 ( P ( x, γ , . . . , γ k , z k +1 )) > . Then any element γ k +1 of an algebraic closure of C ( x ) such that P ( x, γ , . . . , γ k , γ k +1 ) = 0 is a homogeneous element of degree ω k +1 .Proof. The proof is made by induction on k . For k = 0, the result follows directlyfrom the definition. Let now k > , and assume that the result is proved for k − P k ( x, z k ) be a nonzero irreducible ω k -weighted homogeneous polynomial suchthat P k ( x, γ k ) = 0. We set e P ( x, z , . . . , z k − , z k +1 ) := Res z k ( P k ( x, z k ) , P ( x, z , . . . , z k +1 ))where Res z k denotes the resultant of two polynomials seen as polynomials in the in-determinate z k . Then e P ( x, z , . . . , z k − , z k +1 ) is a ( ω , . . . , ω k − , ω k +1 )-weighted ho-mogeneous polynomial, and we have e P ( x, γ , . . . , γ k − , γ k +1 ) = 0. Because P k ( x, z k )is irreducible, gcd( P k ( x, z k ) , P ( x, γ , . . . , γ k − , z k , z k +1 )) is either 1 or P k ( x, z k ), butdeg z k +1 ( P ( x, γ , . . . , γ k , z k +1 )) > , thus P k ( x, z k ) does not divide P ( x, γ , . . . , γ k − , z k , z k +1 ), implying that P k ( x, z k )and P ( x, γ , . . . , γ k − , z k , z k +1 ) are coprime and e P ( x, γ , . . . , γ k − , z k +1 ) is nonzero.We conclude by induction. (cid:3) Corollary 5.21 (Compatibility between homogenous elements) . If γ and γ arehomogeneous elements, respectively of degrees ω and ω , then i) for every q ∈ Q > , γ q is a homogeneous element of degree qω , ii) γ γ is a homogeneous element of degree ω + ω , iii) if ω = ω , γ + γ is a homogeneous element of degree ω .Proof. For i = 1 and 2, let us denote by P i ( x, z i ) a nonzero ω i -weighted homo-geneous polynomial with P i ( x, γ i ) = 0. In order to prove the corollary, we applyLemma 5.20 to P ( x, z , z ) = z a − z b if q = a/b in case i), P ( x, z , z , z ) = z − z z in case ii), and P ( x, z , z , z ) = z − z − z in case iii). (cid:3) We now turn to the proof of a key result in order to reduce the study frommultiple homogeneous elements to a single one:
Lemma 5.22 (Existence of a primitive integral homogeneous element) . Let γ =( γ , . . . , γ r ) be homogeneous elements. Then there exists an integral homogeneouselement γ such that b V ν,γ ⊂ b V ν,γ . Proof.
For simplicity we assume r = 2. The general case is proven in a similar way.Let us denote by ω i the degree of γ i , and write ω i = a i b i where a i , b i ∈ N . We set γ ′ := γ /a b and γ ′ := γ /a b . Therefore γ ′ and γ ′ are homogeneous elements ofthe same degree ω = b b . By the Primitive Element Theorem, there exists c ∈ C such that C ( x )( γ ′ , γ ′ ) = C ( x )( γ ′ + cγ ′ ) . Therefore b V ν,γ ′ ,γ ′ = b V ν,γ ′ + cγ ′ . Since b V ν,γ ,γ ⊂ b V ν,γ ′ ,γ ′ , this proves the existence ofa homogeneous element γ such that b V ν,γ ,γ ⊂ b V ν,γ . Thus, we only need to prove that γ may be chosen to be an integral homogeneouselement. Let us assume that(12) c ( x ) γ d + c ( x ) γ d − + · · · + c d ( x ) = 0where the c k ( x ) are homogeneous polynomials of degree ωk + p , where ω ∈ Q > and p ∈ N . Let γ ′ := c ( x ) γ . Then, by multiplying (12) by c ( x ) d − , we have γ ′ d + c ( x ) γ ′ d − + · · · + c k ( x ) c ( x ) k − γ ′ d − k + · · · + c ( x ) d − c d ( x ) = 0 . This shows that γ ′ is a integral homogeneous element of degree p + ω . This provesthe lemma since b V ν,γ = b V ν,γ ′ . (cid:3) Finally, we are ready to prove Theorem 5.7. We divide the proof in two mainsteps (which combined immediately yield Theorem 5.7), which are interesting ontheir own. The first step shows that we can concentrate our discussion to the rings b V ν,γ instead of P ν J x, γ K . The proof of this first step follows from arguments in thesame spirit of Tougeron’s implicit Function theorem [To72, Chapter 3, Theorem3.2]: Proposition 5.23 (Newton-Puiseux-Eisenstein: Step I) . Let P ( x, y ) ∈ C J x K [ y ] bea monic polynomial. Suppose that there exists an integral homogeneous element γ such that P ( x, y ) factors as a product of degree polynomials in y with coefficientsin b V ν,γ . Then there exists an homogeneous polynomial h such that P ( x, y ) factorsas a product of degree polynomials in y with coefficients in P h J x, γ K .Proof. Let ξ be a root of the polynomial P ( x, y ); by hypothesis ξ ∈ b V ν,γ , so it hasthe following form: ξ = d X i =1 A i γ i , where A i ∈ Frac (cid:16) b V ν (cid:17) and ν ( A i γ i ) > . Let us denote by γ =: γ , · · · , γ d the conjugates of γ over C ( x ). Then d is the degreeof the minimal polynomial of γ over C ( x ). We denote by M the Vandermondematrix whose coefficients are the γ ji , for 1 i, j d . Note that: ξ j := d X i =1 A i γ ij , j = 1 , . . . , d ABRIELOV’S RANK THEOREM 39 are also roots of P ( x, y ). We now have that: M · A ... A d = ξ ... ξ d , therefore A ... A d = M − ξ ... ξ d . Because the entries of M are algebraic over C ( x ), the entries of M − are alsoalgebraic over C ( x ). Thus the A i are algebraic over C (( x )). Now, we claim thatif A ∈ b V ν is algebraic over C (( x )), then there exists h (depending on A ) such that A ∈ P h (( x )). The Proposition easily follows from the Claim, by using the fact that: P h J x, γ K ∪ P h J x, γ K ⊂ P h J x, γ K where h = lcm( h , h ) , and noting that there are a finite number of roots ξ of P , each one of them with afinite number of coefficients A i . We now turn to the proof of the Claim.Let A ∈ Frac (cid:16) b V ν (cid:17) be algebraic over C (( x )). Note that if x A ∈ P h (( x )), then A ∈ P x h (( x )) and, if A is algebraic, then x A is algebraic too. So, up to multiplying A by a large power of x , we may assume that ν ( A ) >
0. We write A = P i> a i ( x ) b i ( x ) ,where a i , b i are homogeneous polynomials such that ν ( a i ) − ν ( b i ) = i , and wedenote by P ( x, z ) the minimal polynomial of A . Now, we replace x i by tx i forevery i . Therefore, Q ( z ) := P ( tx, z ) (where we leave the dependency in x and t implicit) is separable and Q ( B ) = 0, where B := P i> A i ( x ) t i . Note that Q maynot be irreducible over C (( t, x ))[ z ] but it is separable because P is, as an irreduciblepolynomial over a field of characteristic zero. We set e := ord t (cid:16) ∂Q∂z ( B ) (cid:17) > t ( · ) denotes the order of a series with respect to the variable t . Note that ∂Q∂z ( B ) = 0, since Q is separable. We set B := P i e +1 A i ( x ) t i and y = t e +1 v . ByTaylor expansions in z centered at B and B , we have Q ( t e +1 v + B ) = Q ( B ) + ∂Q∂z ( B ) t e +1 v + f Q ( t, x, v ) t e +2 v , and Q ( t e +1 v + B ) = ∂Q∂z ( B ) t e +1 v + f Q ( t, x, v ) t e +2 v , where f Q , f Q ∈ C (( x, t ))[ v ]. Writing B = B + t e +1 P i> e +1 a i ( x ) b I ( x ) t i − e − , this givesord t (cid:18) ∂Q∂z ( B ) (cid:19) = ord t (cid:18) ∂Q∂z ( B ) (cid:19) = e, and ord t ( Q ( B )) > e + 1.Now, note that every term of B is of the form a i b i t i , where a i ( x ) , b i ( x ) are ho-mogeneous polynomials and ν ( a i ) − ν ( b i ) = i . Furthermore, since Q ∈ C (( tx ))[ z ],every term of Q ( B ) is the form f i g i t i , where, again, f i ( x ) , g i ( x ) are homogeneouspolynomials and ν ( f i ) − ν ( g i ) = i . Finally, since B is a finite sum with homoge-neous denominators and Q ∈ C (( xt ))[ z ], there exists an homogeneous polynomial b ( x ) such that b ( x ) Q ( t e +1 v + B ) ∈ C J x, t K [ v ]. Therefore, dividing the equation b ( x ) Q ( t e +1 v + B ) = 0 by t e +1 , we obtain(13) R ( t, x ) + R ( t, x ) v + e R ( t, x, v ) v = 0 , with ord t ( R ) = 0, so we can write R ( t, x ) = g ( x ) + t e R ( t, x ) where g ( x ) is anonzero homogeneous polynomial of degree > e + 1. Moreover v = X i> e +1 A i ( x ) t i − e − is a solution of (13) and, thus, t divides R ( t, x ), that is R ( t, x ) = t e R ( t, x ). Weset t = g ( x ) s and v = g ( x ) w. Then, dividing (13) by g ( x ) we get(14) s e R ( g ( x ) s, x ) + h g ( x ) s e R ( g ( x ) s, x ) i w + e R ( g ( x ) s, x, g ( x ) w ) w = 0 . By the Implicit Function Theorem, (14) has a unique solution w ( s, x ) ∈ C J s, x K , w ( s, x ) = P i ∈ N w i ( x ) s i . Therefore X i ∈ N a i ( x ) b i ( x ) t i = X i e +1 a i ( x ) b i ( x ) t i + X i ∈ N w i ( x ) g ( x ) g ( x ) i t i + e +1 . Hence, if h denotes the product of g with the b i for i e + 1, we have that A ( x ) ∈ P h J x K . This finishes the proof. (cid:3) We now turn to the second step of the proof of Theorem 5.7, which is actually amore general statement than Theorem 5.7, and easier to prove by induction:
Proposition 5.24 (Newton-Puiseux-Eisenstein: Step II) . Let γ be an integralhomogeneous element and P ( x, y ) ∈ b V ν,γ [ y ] be a monic polynomial. Then thereexists an integral homogeneous element γ ′ such that P ( x, y ) factors as a product ofdegree polynomials in y with coefficients in b V ν,γ ′ .Proof. We prove the Proposition by induction on the degree deg y ( P ) = d . Fixed γ , note that the result is trivial when d = 1, and suppose that the Propositionis proved for every homogeneous element γ and every polynomial in b V ν,γ [ y ] withdegree d ′ < d . We fix an homogeneous element γ , and let P ( x, y ) ∈ b V ν,γ [ y ] be amonic polynomial of degree d . Let us write P ( y ) = y d + a y d − + · · · + a d , a i ∈ b V ν,γ , i = 1 , . . . , d. Up to replacing y by y − a /d , we can assume that a = 0. Let k ∈ { , . . . , d } besuch that(15) ν ( a k ) k ν ( a k ) k ∀ k ∈ { , . . . , d } . Let γ be such that γ k − in ν ( a k ) = 0. By Corollary 5.21, γ is a homogeneouselement. By (15), we can write a k = b k γ k for every k ∈ { , . . . , d } , where b k ∈ b V ν,γ,γ . We remark that P ( γ y ) = γ d ( y d + b y d − + · · · + b d ) . We set Q ( y ) = y d + b y d − + · · · + b d . We denote by m , the maximal ideal of b V ν,γ,γ , i.e. the set of elements a such that ν ( a ) >
0, and we denote by Q ( y ) theimage of Q ( y ) in b V ν,γ,γ / m [ y ] ≃ C [ y ]. Then Q ( y ) is not of the form ( y − c ) d since b = 0 and b k / ∈ m . Therefore we may factor Q ( y ) as a product R ( y ) R ( y ), wherethe R i ( y ) are monic polynomials with complex coefficients, and R ( y ) and R ( y )are coprime. By Hensel’s Lemma, this factorization of Q ( y ) lifts as a factorization ABRIELOV’S RANK THEOREM 41 of Q ( y ): Q ( y ) = Q ( y ) Q ( y ) where Q i ( y ) mod m = R i ( y ) for i = 1, 2. Now,by Lemma 5.22, we know that there exists an homogeneous element γ ′ such that Q ( y ) and Q ( y ) are polynomials with coefficients in b V ν,γ ′ , and each one of themhas degree d i < d for i = 1 ,
2. We conclude by induction and by Lemma 5.22 onceagain. (cid:3)
Remark 5.25.
The classical Newton-Puiseux Theorem asserts that a monic poly-nomial P ( y ) with coefficients in k J x K , where x is a single indeterminate and k is acharacteristic zero field, has its roots in k ′ J x /d K for some d ∈ N ∗ and k −→ k ′ afinite field extension.Fix i ∈ { , . . . , n } and let a ( x ), b ( x ) be nonzero homogeneous polynomials. Wehave a ( x ) b ( x ) = x deg( a ) − deg( b ) i a (cid:16) x x i , . . . , x n x i (cid:17) b (cid:16) x x i , . . . , x n x i (cid:17) . This proves that b V ν is isomorphic to K J x i K where K ≃ C (cid:16) x x i , . . . , x n x i (cid:17) .Moreover, if γ is an integral homogeneous element, we can consider the coeffi-cients of its minimal polynomial as elements of K J x K by the previous remark. Inthis case, it is straightforward to check that γ is identified with cx /d u ( x ) where c is algebraic over K , d ∈ N ∗ and u ( x ) is a unit of K J x K . Therefore, Proposition5.24 is an extension of the classical Newton-Puiseux Theorem for univariate powerseries since P h ⊂ b V ν .On the other hand, the classical Eisenstein Theorem [Ei52] is the following state-ment: Theorem (Eisenstein Theorem [Ei52]) . Given a univariate power series f ( x ) = P k ∈ N f k x k ∈ Q J x K that is algebraic over Z [ x ] , there exists a nonzero natural number b such that ∀ k ∈ N , b k +1 f k ∈ Z . This shows that the f k can be written as a k b k +1 where a k are integers. Therefore,Proposition 5.24 is a natural extension of Eisenstein Theorem to the situation where Z is replaced by C J x K .5.3. On convergent projective rings.
In this subsection we prove Proposition5.12. For this we need a complementary inequality to the following inequality: ∀ p, q ∈ C [ x ] , | pq | | p || q | where | p | is a well chosen norm. The investigation for such complementary inequal-ities is an old problem that goes back to K. Mahler [Ma62]. The complementaryinequality that we need is given in Corollary 5.28 given below. Such an inequalityfollows from results proven in this field. But for the sake of completeness we choseto include a proof of this inequality. Moreover our proof seems to be new and maybe of independent interest. It is based on the Weierstrass division Theorem that iswell know in local analytic geometry. Therefore we begin by defining the needednorms: Definition 5.26.
Let ρ >
0. We denote by C { x } ρ the subring of C { x } of series f ( x ) = P α ∈ N n f α x α such that | f | ρ := X α ∈ N n | f α | ρ | α | < ∞ . The map f ∈ C { x } ρ −→ | f | ρ is an absolute value that makes C { x } ρ a Banachalgebra. Proposition 5.27.
Let h ( x ) ∈ C { x } . Then there is ρ > and C > such that forevery a ( x ) ∈ C { x } ρ a ( x ) ∈ h ( x ) C J x K = ⇒ " a ( x ) h ( x ) ∈ C { x } ρ and (cid:12)(cid:12)(cid:12)(cid:12) a ( x ) h ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ρ C | a ( x ) | ρ Proof.
The proof is essentially the proof given in [To72] of the Weierstrass Divisiontheorem. First we fix ρ > h ( x ) ∈ C { x } ρ . If h ( x ) isa unit, up to shrinking ρ , the result is straightforward.If ord( h ( x )) = 1, there is an analytic diffeomorphism ϕ : C { x } −→ C { x } sending h ( x ) onto x . For every ρ , such a diffeomorphism sends C { x } ρ onto C { x } ρ ′ for some ρ ′ >
0. Therefore we may assume that h ( x ) = x , and in this case the result easilyfollows, with C = ρ − .Therefore, we assume that ord( h ( x )) >
2. Up to a linear change of coordinates,we may assume that h ( x ) is x n -regular of order d >
2, that is, h (0 , x n ) = x dn u ( x n )with u (0) = 0. Note that we may identify the C -vector space of polynomials r ( x ) ∈ C { x ′ } ρ [ x n ] of degree at most d in x n with ( C { x ′ } ρ ) d . We now define thefollowing C -linear maps: L , L : C { x } ρ × ( C { x ′ } ρ ) d −→ C { x } ρ by L ( q, r ) = qh + r and L ( q, r ) = qx dn + r. Set L = L − L . We remark that L is a linear isomorphism (every series a ( x )can be written in a unique way as a ( x ) = x dn a ( x )+ a ( x ) where a ( x ) ∈ C { x ′ } ρ [ x n ]has degree < d in x n ). We claim that L and L − are continuous. Indeed, since qx dn and r formal expansion have disjoint support, we have | L ( q, r ) | ρ = | qx dn + r | ρ = | q | ρ ρ d + | r | ρ C ( ρ ) max {| q | ρ , | r | ρ } , | qx dn + r | ρ = | q | ρ ρ d + | r | ρ > min { ρ, } max {| q | ρ , | r | ρ } for some positive constant C ( ρ ) depending on ρ .Now, we claim that L L − is a continuous linear map of norm < ρ chosensmall enough. Indeed, we have that L ( q, r ) = q ( x dn − h ) and, therefore | L ( q, r ) | ρ | x dn − h | ρ | q | ρ , hence, for every a ( x ) ∈ C { x } ρ we have | L L − ( a ( x )) | ρ | x dn − h | ρ min { ρ, } | a ( x ) | ρ . But for ρ small enough, we have | x dn − h | ρ cρ since ord( x dn − h ( x )) >
2, for someconstant c >
0. Therefore, the linear map11 − L L − : C { x } ρ −→ C { x } ρ ABRIELOV’S RANK THEOREM 43 is an isomorphism of Banach algebras. Thus L = (11 − L L − ) L is an isomorphism of Banach algebra if ρ is chosen small enough. Therefore, thereis a constant C , such that for every a ( x ) ∈ C { x } ρ , there is a unique couple ( q, r ) ∈ C { x } ρ × ( C { x ′ } ρ ) d , such that a ( x ) = q ( x ) h ( x ) + r ( x ) and max {| q | ρ , | r | ρ } C | a ( x ) | ρ . Now, the fact that a ( x ) ∈ h ( x ) C J x K is equivalent to r = 0, and then a/h = q , whosenorm is bounded by C | a ( x ) | ρ . (cid:3) Corollary 5.28.
Let h ( x ) be a homogeneous polynomial. Then, there is a constant K > such that, for every homogeneous polynomial b ( x ) : | h | | b | K | hb | . Proof.
Let ρ satisfy the previous proposition for h ( x ). Then, with a ( x ) = b ( x ) h ( x ),we get | b ( x ) | ρ C | h ( x ) b ( x ) | ρ Since h and b are homogeneous, this gives ρ deg( b ) | b ( x ) | Cρ deg( b )+deg( h ) | h ( x ) b ( x ) | , and | h ( x ) | | b ( x ) | C | h ( x ) | ρ deg( h ) | h ( x ) b ( x ) | . (cid:3) We are ready to complete the goal of this section:
Proof of Proposition 5.12.
First, let us remark that P k ∈ N a k ∈ C { x } , where the a k are homogeneous polynomials of degree k , if and only if P k a k ∈ C { x } ρ , for some ρ >
0, that is, P k ∈ N | a k | ρ < ∞ .Let h and h be two homogeneous elements such that A := X k ∈ N a ,k h α k + β = X k ∈ N a ,k h α k + β and assume that P k a ,k ∈ C { x } . We have X k a ,k h α k + β = X k a ,k h α k + β ( h h ) α k + β . Let ρ >
0. We have | a ,k h α k + β | ρ | a ,k | ρ | h | α k + β ρ = | a ,k | | h | α k + β ρ deg( a ,k )+deg( h )( α k + β ) But deg( a ,k ) αk + β for some α , β ∈ N . Thus we have | a ,k h α k + β | ρ | a ,k | ( ρ α +deg( h ) α | h | α ) k ρ β C for some constant C >
0. Assume that P k a ,k ∈ C { x } ρ ′ for ρ ′ >
0. Then, for ρ > ρ α +deg( h ) α < | h | − α ρ ′ , we have X k ∈ N | a ,k h α k + β | ρ ρ β C X k ∈ N | a ,k | ρ ′ k = ρ β C X k | a ,k | ρ ′ < ∞ . In the same way, since a ,k h α k + β ( h h ) α k + β = a ,k h α k + β ( h h ) max { ,α − α } k +max { ,β − β } ( h h ) max { α ,α } k +max { β ,β } we may assume that A = X k ∈ N a ,k ( h h ) αk + β = X k ∈ N a ,k h αk + β and P k a ,k ∈ C { x } . Hence for every k ∈ N , we have a ,k = a ,k h αk + β . Thus, by Corollary 5.28, there is a constant
K > ∀ k ∈ N , | a ,k h αk + β − | K | h | | a ,k | , and, by induction, ∀ k ∈ N , | a ,k | K αk + β | h | αk + β | a ,k | . Therefore, if P k a ,k ∈ C { x } ρ ′ for ρ ′ >
0, we have that P k a ,k ∈ C { x } ρ for every ρ > ρ < | h | α ρ ′ K α . (cid:3) Remark 5.29.
In [Ma62], Mahler proved the following inequality: ∀ h, b ∈ C [ x ] , | hb | deg( h )+deg( b ) | p | | q | . Such an inequality has been first proved by Gel’fond for polynomials in one inde-terminate [Ge52]. We could have used this inequality in order to prove Proposition5.12.The inequality of Mahler has the advantage of being effective and uniform inboth h and b . On the other hand, the inequality given in Corollary 5.28 shows thatthe multiplicative factor can be chosen to be independent of b when h is fixed.5.4. Semi-global formal extension.
The proof of Theorem 5.15 is done in twosteps. First, assuming that P ∈ C { x } [ y ] is a convergent polynomial, we can providean elementary proof based on analytic continuation and on the classical Abhyankar-Jung Theorem. More precisely: Proposition 5.30 (Semi-global formal extension: the analytic case) . Let P ( x, y ) ∈ C { x } [ y ] be a monic reduced polynomial, and let P = Q si =1 Q i be the factorizationof P given in Corollary 5.8. Suppose that n = 2 and let σ : ( N, F ) −→ ( C , bea sequence of point blowing ups such that, at every point b ∈ F (1) r , the pulled-backdiscriminant σ ∗ b (∆ P ) is analytically monomial. Then, for every point b ∈ F (1) r ,each polynomial Q i extends formally at b . Furthermore, the extension is compatiblewith the factorization of P , that is Q ri =1 σ ∗ b ( Q i ) = σ ∗ b ( P ) .Proof. Given a point b ∈ F (1) r which is not in the strict transform of ( h = 0), northe intersection of exceptional divisors, then each σ ∗ b ( Q i ) is a monic formal factorsof the convergent monic polynomial σ ∗ b ( P ). We conclude therefore that each Q i isconvergent. In particular, note that Q i ∈ P h { x } [ y ] by Lemma 5.13. We now studythe points b ∈ F (1) r which are either in the strict transform of ( h = 0), or in anintersection of exceptional divisors. ABRIELOV’S RANK THEOREM 45
By combining the normal forms given in Remark 5.6(2), up to a linear changeof coordinates, there exists a coordinate system ( v, w ) centered at b and c ∈ Z > such that: x = v · w c , x = v · w c +1 Now, by hypothesis, we know that ∆ P b := σ ∗ b (∆ P ) is a analytically monomial(adapted to the exceptional divisor). More precisely, up to an analytic change ofcoordinates adapted to the exceptional divisor, that is: v = e v, w = ϕ ( e v, e w )we can assume that ∆ P b = e v k e w l u , where u is a unit (note that this is always thecase if b is in the intersection of two exceptional divisors, when we necessarily have w = e w ). Then, by Abhyankar-Jung Theorem, we may write P b = d Y i =1 ( y − ξ i ( e v d ! , e w d ! ))where the ξ i ( e v d ! , e w d ! ) ∈ C { e v d ! , e w d ! } . Now, there exists a connected punctureddisc D ⊂ F (1) r centred at b (in particular, b / ∈ D ) such that every point c ∈ D doesnot belong to the strict tranform of ( h = 0), nor to the intersection of exceptionaldivisors. It follows that σ ∗ c ( Q i ) is a well-defined analytic function. Furthermore, upto shrinking D , for every i = 1 , . . . , r , there exists a subset J i ( c ) ⊂ { , . . . , d } suchthat: σ ∗ c ( Q i ) = Y j ∈ J i ( c ) ( y − ξ j ( e v d ! , ( e w − w ) d ! )) , where b = (0 , w ). Since D is a connected set, we conclude that J i ( c ) is independentof the point c ∈ D , and we denote it by J i . There exists, therefore, an element q ∈ C { e v /d ! , e w /d ! } [ y ] which is equal to σ ∗ c ( Q i ) at every point c ∈ D . On the onehand, it follows, from analytic continuation, that σ ∗ b ( Q i ) is equal to q as a powerseries in C { e v /d ! , e w /d ! } [ y ]. Now, on the other hand, if A ∈ P h J x K is a coefficient of Q i then: A b := σ ∗ b ( A ) = X k > a k (1 , ϕ ( e v, e w )) h (1 , ϕ ( e v, e w )) αk + β e v k ϕ ( e v, e w ) ck , and A b is invariant by the action of the ( d !)-th roots of unity ( ζ , ζ ) ( ζ e v, ζ e w ),which implies that q is also invariant by these actions. We conclude that q =: σ ∗ b ( Q i )is a power series in C { e v, e w } [ y ], finishing the proof. (cid:3) Unfortunately, there is no notion of analytic continuation in the case of formalpower series, so the proof does not extend in a trivial way to the formal case.Following Gabrielov’s original idea [Ga73], we address this extra technical issue viathe Artin Approximation Theorem:
Theorem 5.31 (Artin Approximation Theorem [Ar68]) . Let G ( x, y ) be a nonzerofunction in C { x } [ y ] , where x = ( x , . . . , x n ) and y = ( y , . . . , y m ) , and considerthe equation G ( x, y ) = 0 . Suppose that there exists formal power series ˆ y ( x ) =(ˆ y ( x ) , . . . , ˆ y m ( x )) ∈ ( C J x K ) m such that: b G ( x, ˆ y ( x )) ≡ Then there exist convergent power series y ( ι ) ( x ) = ( y ( ι )1 ( x ) , . . . , y ( ι ) m ( x )) ∈ ( C { x } ) m for every ι ∈ N , such that: G ( x, y ( ι ) ( x )) ≡ and for every k, ˆ y k ( x ) − y ( ι ) k ( x ) ∈ ( x ) ι . Before turning to the proof of Theorem 5.15, we need to study conditions underwhich a well-chosen approximation of a polynomial yields an approximation oftheir roots and factors. We start by setting, once and for all, the extension of thevaluation ν which we are interested in working with: Definition 5.32.
Let V be a ring equipped with a valuation ν : V r { } −→ R .Let P ( y ) ∈ V [ y ], P ( y ) = P dk =0 a i y i , a i ∈ V for every i . We define ν ( P ( y )) = min { ν ( a ) , . . . , ν ( a d ) } . We now turn to the proof of two preliminary results:
Lemma 5.33 (Root-approximation) . Let K be an algebraically closed field equippedwith a valuation ν : K r { } −→ R . We denote by V its valuation ring. Let P ( y ) ∈ V [ y ] be a reduced monic polynomial of degree d in y , and ξ , . . . , ξ d be itsroots in K . Let Q ( y ) ∈ V [ y ] be a monic polynomial such that ν ( P − Q ) > d · ν (∆ P ) . Then, for every i = 1 , . . . , d , Q ( y ) has a unique root ξ ′ i in K such that ν ( ξ i − ξ ′ i ) > ν ( P − Q ) d . Note that the unicity implies that, in these conditions, Q is necessarily reduced. Proof.
We start by remarking that, if the degree deg y ( Q ) is different from d , then ν ( P − Q ) = 0 since these are monic polynomials, and the hypothesis of the Lemmaare not satisfied. We therefore have that deg y ( Q ) = d , and we denote by ξ ′ , . . . , ξ ′ d the roots of Q ( y ) counted with multiplicity (hence we may have ξ ′ i = ξ ′ j for some i = j ). Since P and Q are monic polynomials, we have ν ( ξ i ), ν ( ξ ′ j ) > i and j . We write P ( y ) = y d + a y d − + · · · + a d , Q ( y ) = y d + b y d − + · · · + b d . Let us fix i ∈ { , . . . , d } . We have d Y j =1 ( ξ i − ξ ′ j ) = Q ( ξ i ) = Q ( ξ i ) − P ( ξ i ) = d X k =1 ( b k − a k ) ξ d − ki , and, since ν ( ξ i ) >
0, we get: ν d Y j =1 ( ξ i − ξ ′ j ) > d min k =1 { ν ( a k − b k ) } = ν ( P − Q ) . Therefore, there is (at least) one integer j ( i ) such that(16) ν ( ξ i − ξ ′ j ( i ) ) > d ν ( P ( y ) − Q ( y )) > ν (∆ P ) > max l k { ν ( ξ k − ξ l ) } . Now let m ∈ { , . . . , d } , m = i . Then, by (16), we get ν ( ξ m − ξ ′ j ( i ) ) = ν ( ξ m − ξ i + ξ i − ξ ′ j ( i ) ) = ν ( ξ m − ξ i ) max l k { ν ( ξ k − ξ l ) } . ABRIELOV’S RANK THEOREM 47
Therefore j ( m ) = j ( i ) whenever i = m . This gives the unicity of ξ ′ i and concludesthe proof. (cid:3) Proposition 5.34 (Factor-approximation) . Let P ( y ) ∈ b V ν [ y ] be a reduced monicpolynomial of degree d in y . We write P ( y ) = P ( y ) · · · P s ( y ) where the P i ( y ) are irreducible monic polynomials of b V ν [ y ] . Let Q ( y ) ∈ b V ν [ y ] be amonic polynomial such that ν ( P ( y ) − Q ( y )) > dν (∆ P ) . Then Q ( y ) can be written as a product of irreducible monic polynomials Q i [ y ] ∈ b V ν [ y ] : Q ( y ) = Q ( y ) · · · Q s ( y ) , such that ν ( P i − Q i ) > ν ( P − Q ) d i = 1 , . . . , s. Proof.
Again, if the degree deg y ( Q ) is different from d , then ν ( P − Q ) = 0 sincethese are monic polynomials, and the hypothesis of the Proposition are not satisfied.We therefore have deg y ( Q ) = d . We denote by ξ , . . . , ξ d the distinct roots of P ( y ),belonging to b V ν [ γ ] for some homogeneous element γ , and by ξ ′ , . . . , ξ ′ d the roots of Q ( y ) counted with multiplicity (hence we may have ξ ′ i = ξ ′ j for some i = j ).By Lemma 5.33, for every i ∈ { , . . . , d } , there is a unique j ( i ) such that ν ( ξ i − ξ ′ j ( i ) ) > ν ( P − Q ) d , and, up to renumbering, we can suppose that j ( i ) = i . Now, fix i ∈ { , . . . , d } .There exists p ∈ N (where p depends on i ) and σ = Id, σ , . . . , σ p Frac( b V ν )-automorphisms of K ν , such that ξ i = σ ( ξ i ), . . . , σ p ( ξ i ) are the distinct conjugatesof ξ i over b V ν . Then Q pk =1 ( y − σ k ( ξ i )) is an irreducible factor of P ( y ) in b V ν [ y ].Now let σ be any Frac( b V ν )-automorphism of K ν . Because σ ( ξ i ) is a conjugateof ξ i , there is an integer l such that σ ( ξ i ) = σ l ( ξ i ). Moreover we have ν ( σ ( ξ ′ i ) − σ ( ξ i )) = ν ( ξ ′ i − ξ i ) > ν ( P − Q ) d . Therefore ν ( σ ( ξ ′ i ) − σ l ( ξ i )) = ν ( σ ( ξ ′ i ) − σ ( ξ i )) = ν ( ξ ′ i − ξ i ) == ν ( σ l ( ξ ′ i ) − σ l ( ξ i )) > ν ( P − Q ) d , and we conclude that σ ( ξ ′ i ) = σ l ( ξ ′ i ). This proves that σ ( ξ ′ i ), . . . , σ p ( ξ ′ i ) areexactly the (distinct) conjugates of ξ ′ i over Frac( b V ν ). In particular the polynomial Q pk =1 ( y − σ k ( ξ ′ i )) is an irreducible monic factor of Q ( y ) in b V ν [ y ] such that ν p Y k =1 ( y − σ k ( ξ i )) − p Y k =1 ( y − σ k ( ξ ′ i )) ! > ν ( P − Q ) d , as we wanted to prove. (cid:3) We are now ready to prove the main result of this section:
Proof of Theorem 5.15.
The proof is done by approximating the formal polynomial P ( x, y ) by a suitable sequence of analytic polynomials ( P ( ι ) ) ι ∈ N , which satisfies allhypothesis of Proposition 5.30, and arguing via Proposition 5.34. Indeed, we startwriting: P = P ( x ) + · · · + P d − x ) y d − + y d , ∆ P = ∆ k · · · ∆ k e e , where the ∆ i are distinct irreducible formal polynomials and k i > i =1 . . . , e . We consider the universal discriminant polynomial ∆ d of degree d :=deg y ( P ), that is, ∆ d ( P , . . . , P d − ) is the discriminant of the polynomial P + · · · + P d − y d − + y d . We apply the Artin Approximation Theorem to the equation:∆ d ( y , . . . , y d − ) = z k · · · z k e e with respect to the formal solution y i = P i ( x ) and z i = ∆ i ( x ). Therefore, for every ι ∈ N , there exist P ( ι )0 ( x ) , . . . , P ( ι ) d − ( x ) ∈ C { x } such that P ( ι ) j ( x ) − P j ( x ) ∈ ( x ) ι , j = 0 , . . . , d − P ( ι ) ( x, y ) := P ( ι )0 ( x ) + · · · + P ( ι ) d − ( x ) y d − + y d which has a discriminant, by construction, of the form∆ P ( ι ) = (∆ ( ι )1 ) k · · · (∆ ( ι ) e ) k e , where ∆ ( ι ) j − ∆ j ∈ ( x ) ι , j = 1 , . . . , e. Note that, by Theorem 5.7, these polynomials admit a factorization P ( ι ) = Q s ( ι ) i =1 Q ( ι ) i in P h ( ι ) J x K [ y ] ⊂ b V ν [ y ]. By Proposition 5.34, for ι big enough, the number of factors s ( ι ) is constant equal to s and P ( ι ) is a reduced in b V ν [ y ].In order to apply Proposition 5.30, it remains to verify that there exists ι ∈ N such that, for every ι > ι and for every b ∈ F (1) r , the discriminant σ ∗ b (∆ P ( ι ) ) isanalytically monomial. Indeed, fix a point b ∈ F (1) r . By Remark 5.6(2), there existsa coordinate system ( v, w ) centered at b and adapted to F given by: x = vw c , x = vw c +1 with c ∈ Z > . In particular, we note that:(17) σ ∗ b (∆ ( ι ) j ) − b σ ∗ b (∆ j ) ∈ ( vw c ) ι , j = 1 , . . . , e. We now divide in three cases in order to prove that σ ∗ b (∆ ( ι ) j ) is analytically monomial(for ι big enough) depending on the nature of b : Case I:
Suppose that b is in the intersection of two exceptional divisors. In thiscase, c > b σ ∗ b (∆ j ) = v α j w β j b u j , where α j , β j > b u j (0) = 0, for every j =1 , . . . , e . It follows from (17) that if ι > max { α j , β j } , then σ ∗ b (∆ ( ι ) j ) is analyticallymonomial. We easily conclude that σ b (∆ P ( ι ) ) is analytically monomial for ι > max ej =1 { α j , β j } . Case II:
Suppose that b is not in the intersection of two exceptional divisors, nor inthe strict transform of ∆ P . In this case, c = 0 and ∆ j = v α j b u j , where α j > b u j (0) = 0, for every j = 1 , . . . , e . It follows from (17) that if ι > α j , then σ ∗ b (∆ ( ι ) j ) isanalytically monomial. We easily conclude that σ ∗ b (∆ P ( ι ) ) is analytically monomialfor ι > max ej =1 { α j } . ABRIELOV’S RANK THEOREM 49
Case III:
Suppose that b is in the strict transform of ∆ P . In this case, c = 0. Wenote that, since b σ ∗ b (∆ P ) is formally monomial (adapted to the exceptional divisor),then b σ ∗ b (∆ P ) = v α · g β b u , where α > b u (0) = 0 and g is a formal series with initialterm av + bw , with b = 0. Since each term ∆ j is irreducible and distinct, we concludethat g can only divide one term ∆ j , say ∆ . Therefore, we can write ∆ = v α g · b u and ∆ j = v α j b u j , where α j > b u j (0) = 0, for every j = 1 , . . . , e , and b u (0) = 1.Now, it follows from (17) that if ι > α j , then σ ∗ b (∆ ( ι ) j ) is analytically monomialfor j = 2 , . . . , e . Next, it follows from (17) that if ι > α , then σ ∗ b (∆ ( ι ) j ) = v α g ( ι ) ,where ∂ w g ( ι ) (0) = b = 0. It follows from the implicit function Theorem, therefore,that σ ∗ b (∆ ( ι ) j ) is analytically monomial and adapted to the exceptional divisor. Weeasily conclude that σ ∗ b (∆ P ( ι ) ) is analytically monomial for ι > α .Therefore, since F (1) r is compact and ∆ ( ι ) j are convergent power series for every ι and j = 1 , . . . , e , we conclude that there exists ι ∈ N such that, for every ι > ι and for every b ∈ F (1) r , the discriminant σ ∗ b (∆ P ( ι ) ) is analytically monomial. ByProposition 5.30, the factors Q ( ι ) i admits a formal extension to every point b ∈ F (1) r (which are compatible with the factorization of P ( ι ) b ). Next, by Proposition 5.34,the sequence ( Q ( ι ) j ) ι converges to Q j with respect to the usual ν -adic topology in b V ν , for every j = 1 , . . . , s .Now, let j ∈ { , . . . , s } be fixed, A be a coefficient of Q j and A ( ι ) be the corre-sponding coefficient of Q ( ι ) j . Then we have σ ∗ b ( A ( ι ) ) = X k > a k,ι (1 , w ) h α ι k + β ι ι (1 , w ) ( w c v ) k and, since A ( ι ) extends analytically at b , there exist polynomials b k,ι ( w ) such that a k,ι (1 , w ) = b k,ι ( w ) h ι (1 , w ) α ι k + β ι for every k >
0. Since A − A ( ι ) ∈ ( x ) ι , we have σ b ( A ) − σ b ( A ( ι ) ) ∈ ( vw c ) ι . Therefore, if we write A b = X k > a k (1 , w ) h (1 , w ) αk + β ( w c v ) k we have a k (1 , w ) = b k,ι ( w ) h (1 , w ) αk + β for k ι . Since this is true for every ι ,we have that A b extends formally to b , and so does Q i . Finally, this extensionis compatible with the factorization of P because, as it is pointed out in Remark5.6(2), b σ ∗ b : P h J x K [ y ] −→ C ( w ) J v K is a well-defined morphism. (cid:3) Local-to-Semi-global convergence of factors.
The goal of this section isto prove Theorem 5.17. Here, by compact disc we mean a set diffeomorphic tothe closed unit disc in C . Our proof follows the strategy inspired from Tougeron’s[To90, § Lemma 5.35.
Let
C ⊂ C n be an irreducible algebraic curve, and D , D be twocompact discs in the smooth part of C . Then ∃ M > , ∀ P ∈ C [ z , . . . , z n ] , k P k D M deg ( P ) k P k D , where k P k D denotes max z ∈ D | P ( z ) | . The proof given by Tougeron in [To90, Lemme 3.3] is only valid if the compacti-fication
C ⊂ CP n of the curve C is isomorphic to P . In order to treat the general case, we incorporate the idea of the proof of [Iz89, Theorem 3.1], involving Green’sfunctions. Proof.
Let C be the smooth compact curve obtained as the normalization of theclosure of C in CP n . Note that the irreducibility of C is equivalent to that of C .Then the coordinate functions z i and P lift canonically to meromorphic functions z i , P on C . Furthermore, each of these functions has its poles outside of D ∪ D ,where we identify D i with its preimage in C by the normalization.Denote q , . . . , q k the distinct poles of the functions z i , let q be a point in theinterior of D , and for i = 1 , . . . , n , let m i := max j =1 ,...,k mult q i z j , where mult q z = m means that q is a pole of multiplicity m of z . With thesenotations, if q i is a pole of P , it is of multiplicity at most m i · deg( P ). Furthermore,the only possible poles of P are { q , · · · , q k } .Now, denote U := C r { q , q , . . . , q k } . Then, following [La88, Chapter II, § i ∈ { , . . . , k } , there exists a Green’s function for q i , that is, a smoothfunction G i : C r q i −→ R such that, for every i, j , ∆ G i = ∆ G j on U and, on aneighbourhood of q i , the function G i + log | z − q i | can be extended to q i as a smoothfunction. Note that, since lim z → q i log | z − q i | = −∞ , this last condition implies that G i is positive on a neigbourhood of q i . Finally, such functions are uniquely determinedup to additive constants. We can therefore pick the functions G i so that, for every i ∈ { , . . . , k } , G i − G > U r D , by compacity of C . Set G := k X i =1 m i ( G i − G ) , which is harmonic on U , because every function G i − G is harmonic on U , anddenote h = exp( G ). The function | P /h deg( P ) | is subharmonic on U because atevery point of U , it can locally be seen as the square of the module of a holomorphicfunction. Indeed, since G is harmonic on U , for every point p of U , there is aneighbourhood V of p and a harmonic function H on V which is conjugate to G ,that is, H is such that G + iH is holomorphic on V , and we have (cid:12)(cid:12)(cid:12)(cid:12) Ph deg( P ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) P exp( G + iH ) deg( P ) (cid:12)(cid:12)(cid:12)(cid:12) , where exp( G + iH ) deg( P ) is holomorphic. For 1 i k , the function G i can bewritten near the point q i as G i = α i − log | z − q i | for some smooth function α i .Furthemore, because of the bound on its multiplicity, P can be written near q i as β i / ( z − q i ) m i , for some analytic function β i .Therefore, the function | P /h deg( P ) | can be extended to a smooth function on C r D . Moreover, by continuity of thepartial derivatives, the non negativity of the Laplacian of | P /h deg( P ) | extends toevery point q i , for 1 i k , therefore | P /h deg( P ) | extends to a subharmonicfunction on C r D . Then, the maximum principle applied to this function yields k P /h deg( P ) k D k P /h deg( P ) k C r D k P /h deg( P ) k ∂D k P k ∂D k P k D , where the last inequality comes from the fact that h > ∂D . Finally, denoting M = max z ∈ D h ( z ), we get k P k D M deg( P ) k P k D . (cid:3) ABRIELOV’S RANK THEOREM 51
We are now ready to turn to the proof of Theorem 5.17:
Proof of Theorem 5.17.
Let P ∈ C J x K [ y ] be a monic, reduced polynomial, and let σ : ( N, F ) −→ ( C ,
0) be a sequence of point blowing ups, such that the pulled-backdiscriminant b σ ∗ b (∆ P ) is formally monomial at every point b of F (1) r . We considerthe factorization of P into irreducible polynomials of P h J x K [ y ] given by Corollary5.8: P ( x, y ) = s Y i =1 Q i , By hypothesis, there exists b ∈ F (1) r such that b σ ∗ b ( P ) =: P b has a convergentfactor. By Proposition 4.9, we can assume that b is in no other component of theexceptional divisor nor on the strict transform of { h = 0 } . Therefore, by Theorem5.15, we can write P b = s Y i =1 b σ ∗ b ( Q i ) , where b σ ∗ b ( Q i ) ∈ b O b [ y ] . Now, by hypothesis, there is an index i such that b σ ∗ b ( Q i ) admits a convergentfactor. The problem is now to prove that the polynomial Q b := b σ ∗ b ( Q i ) itself isconvergent. For simplicity, in the sequel we denote Q i by Q , and Γ i by Γ (with thenotation of Corollary 5.8). We now use the second equality given by Corollary 5.8.Let γ be a homogeneous integral element of degree ω = p/e such that(18) Q = s Y i =1 ( y − ξ ( x, γ i )) , where ξ ∈ P h J x, γ K , where γ i are distinct roots of the minimal polynomial of γ :(19) Γ( x, z ) = z d + d X i =1 f i ( x ) z d − i . which is a ω -weighted irreducible homogeneous polynomial. In particular, f i ( x )is homogeneous of degree ω · i , f i = 0 if e does not divide i , and e divides d (because f d = 0; otherwise z would divide Γ which is irreducible). We concludethat Γ ∈ C [ x, z e ]. Now, note that by the definition of P h J x, γ K , we can write:(20) ξ ( x, γ i ) = d X j =1 A j ( x ) · γ ji , where A j = X k > k j a k,j ( x ) h α j k + β j ( x ) , with k j + ωj > x = v e , x = v e w, z = v p z in the chart v = 0:Γ( w, z ) = z d + d X i =1 f i (1 , w ) z d − i . Since Γ is irreducible in C [ x, z ], the compact algebraic curve C corresponding to Γin the weighted projective space CP e,e,p ) is irreducible. Claim:
In the chart v = 0, Γ is irreducible as an element of C [ w, z e ]. Proof of the claim.
Indeed, assume that Γ = GH where G , H are non trivial poly-nomials of C [ x, z e ], monic in z . Let us write G = d X i =1 g i ( w ) z d − i + z d where g i ( w ) = 0 if e does not divide i , and e divides d . Then v pd G = d X i =1 v pi g i ( w )( v p z ) d − i + ( v p z ) d . Let i be a multiple of e . Then a monomial of v pi g i ( w ) has the form v pi w k = v p ie e w k = v ( p ie − k ) e ( v e w ) k = x p ie − k x k . Thus G = G ′ ( x, z ) with G ′ ( x, z ) ∈ C ( x )[ z e ]. The same being true for H we have thatΓ factors as a product of two monic polynomials in C ( x )[ z e ]. Since Γ is irreducibleand C [ x ] is a unique factorization domain, by Gauss’s Lemma Γ factors as a productof two monic polynomials of C [ x, z e ], which is not possible by assumption. Thisshows that Γ is irreducible in C [ x, z e ]. (cid:3) Therefore Γ may not be irreducible in C [ w, z ], but its irreducible factors can beobtained from one another by multiplying the second variable by a e -th root ofunity. Denote C := { Γ = 0 } ⊂ C w,z .This implies that: Γ( w, z ) = t Y i =1 Γ i ( w, z )where C i := (Γ i ( w, z ) = 0) are irreducible curves and each of the irreducible factorsΓ i of Γ( w, z ) can be obtained from one another by multiplying the second variableby a e -th root of unity.Now, let ( v, w ) be coordinates at b such that σ b is locally given by ( x , x ) =( v, vw ), as in Remark 5.6(2a). It follows from equation (18) that: Q b = s Y i =1 ( y − ξ i ( v, vw, ˜ γ ))where, by equation (19), ˜ γ is a root of the polynomial: e Γ( v, w, z ) := z d + d X i =1 v ωi f i (1 , w ) z d − i . By comparing the expressions of e Γ and Γ, we conclude that ˜ γ is a root of e Γ( v, w, z )if and only if ˜ γ = v ω γ , where γ is a root of Γ( w, z ).We use again Proposition 4.9 in order to assume that b is not a zero of thediscriminant of Γ( w, z ) with respect to the projection in the w -axis. With thiscondition, the implicit function Theorem implies that there is a compact disc D ′ ⊂{ z = 0 } centered at the origin of C v,w such that, on each one of the connectedcomponents D ′ , . . . , D ′ d of π − ( D ′ ) ⊂ C , z can be written as an analytic functionin w . From now on, γ ∈ C { w } denotes the solution of Γ( w, z ) = 0 on D ′ . ABRIELOV’S RANK THEOREM 53
Up to renumbering, since Q b has a convergent factor, we can assume that b σ ∗ b ( ξ ) ∈ C { v /e , w } . By equation (20), we can write (c.f. the normal form givenin Remark 5.6(2a)): b σ ∗ b ( ξ i ) = d X j =1 b σ ∗ b ( A j ) · v ωj γ ji = d X j =1 v ωj γ ji X k > k j v k a k,j (1 , w ) h (1 , w ) α j k + β j . The coefficient of v ℓ/e , for ℓ ∈ N , in the previous sum is d X j =1 γ j a ℓ/e − jω,j (1 , w ) h (1 , w ) α j ( ℓ/e − jω )+ β j . (where a ℓ/e − jω,j (1 , w ) = 0 if ℓ/e − jω / ∈ N ). Therefore we have b σ ∗ b ( ξ i ) = X ℓ ∈ N v ℓ/e h (1 , w ) δ ℓ P ℓ ( w, γ i ) , where δ ℓ ∈ N is bounded by an affine function in ℓ , and the degree of P ℓ ∈ C [ w, z ]is bounded by an affine function in ℓ .Now, note that if η is a primitive e -th root of unity, then the set { η l γ, l e − } contains a root of each irreducible factor of Γ. Furthermore, sincegcd( p, e ) = 1, given an integer l , ∃ ε ∈ C such that ε p = η and ε e = 1. This impliesthat d X j =1 b σ ∗ b ( A j ) v ωj ( η l γ ) j = b σ ∗ b ( ξ )( εv /e , w ) ∈ C { v /e , w } . which shows that b σ ∗ b ( ξ i ) are convergent whenever γ i is conjugated to γ by a rootof the unit.Now fix an arbitrary ξ i . By the previous argument, up to renumbering, wemay assume that γ and γ i belong to the same irreducible component of C , where b σ ∗ b ( ξ ) ∈ C { v /e , w } is convergent, and let D ′ i and D ′ be the discs correspondingto the roots γ and γ i .The fact that b σ ∗ b ( ξ ) ∈ C { v, w } is equivalent to the existence of A, B > D in the w -axis, containing the origin in its interior, such that (cid:13)(cid:13)(cid:13)(cid:13) P ℓ ( w, γ ) h (1 , w ) δ ( ℓ ) (cid:13)(cid:13)(cid:13)(cid:13) D AB ℓ . Since h (1 , = 0, we can assume that D is such that(21) ∃ C > , ∀ w ∈ D, C | h (1 , w ) | C. In particular we have k P ℓ ( w, γ ) k D AB ℓ C δ ( ℓ ) . Even if it means shrinking D , we may suppose furthermore that D ⊂ D ′ . Denoteby D ⊂ D ′ , . . . , D d ⊂ D ′ d the connected components of π − ( D ). Let D i be theconnected component of π − ( D ) such that ( w, γ i ) ∈ D i . Then k P ℓ ( w, γ i ) k D = k P ℓ ( w, z ) k D i . Since D and D i are in the same irreducible component of C , Lemma5.35 states the existence of a constant M > k P n ( w, z ) k D i M deg( P n ) k P n ( w, z ) k D . Finally, we conclude that k P n ( w, γ i ) k D = k P n ( w, z ) k D i M deg( P ℓ ) AB ℓ C δ ( ℓ ) , and, by (21), (cid:13)(cid:13)(cid:13)(cid:13) P ℓ ( w, γ i ) h (1 , w ) δ ( ℓ ) (cid:13)(cid:13)(cid:13)(cid:13) D M deg( P ℓ ) AB ℓ C δ ( ℓ ) . Therefore, because deg( P ℓ ) and δ ( ℓ ) are bounded by affine functions, we concludethat b σ ( ξ i ) ∈ C { v, w } , for any i . Since the choice of ξ i was arbitrary, we concludethat Q b ∈ C { v, w } [ y ]. Thanks to the assumption that b is not on the stricttransform of { h = 0 } , Lemma 5.13 implies that Q i ∈ P h { x } [ y ].Therefore, we can identify convergent factors of P and of P b , for every b outsideof a discrete subset of F (1) r . In other words, if Q is the maximal convergent factorof P , then Q b is the maximal convergent factor of P b , for every b ∈ F (1) r outside ofa discrete subset of F (1) r . We conclude easily via Proposition 4.9. (cid:3) Applications and variations
In this section we prove the results announced in §§ Proof of Theorem 1.6.
Let us assume that r( ϕ ) = r F ( ϕ ) = r A ( ϕ ). Byreplacing A by A Ker( ϕ ) we may assume that ϕ is injective. Thus r( ϕ ) = r F ( ϕ ) =r A ( ϕ ) = dim( A ). Let f ∈ b ϕ ( b A ) ∩ B . We may assume that f (0) = 0 by replacing f by f − f (0). We define a new morphism ψ : C { x,z } I C { x,z } −→ B , where A = C { x } I and z is a new indeterminate, by ψ ( g ( x, z )) = g ( ϕ ( x ) , . . . , ϕ ( x n ) , f )) for any g ∈ C { x, z } I C { x, z } . Since ψ | A = ϕ , we have r( ψ ) > r( ϕ ). Since f ∈ b ϕ ( b A ), there exist h ∈ b A such that b ϕ ( h ) = f . Thus z − h ∈ Ker( b ψ ). In fact, by the Weierstrass division Theorem, everyelement of Ker( b ψ ) is equal to an element of b A modulo ( z − h ). But, because Ker( b ψ ) ∩ b A = Ker( b ϕ ) = (0), we have that Ker( b ψ ) = ( z − h ). In particular the injection b A −→ C J x,z K I C J x,z K +Ker( b ψ ) is an isomorphism, thus r F ( ψ ) = r F ( ϕ ) = r( ϕ ). Finally,since r F ( ψ ) > r( ψ ), we get r F ( ψ ) = r( ψ ), hence r F ( ψ ) = r A ( ψ ), by Theorem1.4. Therefore, by Proposition 2.1 we have that Ker( ψ ) . b C = Ker( b ψ ). Thus, if f ∈ C { x, z } is a generator of Ker( ψ ), there is a unit u such that f = u ( z − h ). But,by the unicity in the Weierstrass preparation Theorem, we have that u and z − h are convergent, hence h ∈ A . This proves that ϕ is strongly injective.On the other hand, assume that ϕ strongly injective. There exists a finite in-jective morphism C { x } −→ A (by Noether normalization Lemma) and an injectivemorphism of maximal rank B −→ C { y } (by resolution of singularities). Hence, if wedenote by ψ the induced morphism C { x } −→ C { y } , by Proposition 2.2, r( ψ ) = r( ϕ )and r F ( ψ ) = r F ( ϕ ) and ψ is strongly injective. Therefore we are exactly in thesituation of [EH77, Theorem 1.2] that asserts that r( ψ ) = r F ( ψ ) = r A ( ψ ). ABRIELOV’S RANK THEOREM 55
Proof of Theorem 1.7.
In what follows, we prove that ( I ) = ⇒ ( II ) = ⇒ ( III ) = ⇒ ( IV ) = ⇒ ( I ). The Theorem immediately follows because ( I ) isGabrielov’s rank Theorem 1.4.( I ) = ⇒ ( II ) By replacing A by A Ker( ϕ ) we may assume that ϕ is injective, thusr( ϕ ) = dim( A ) = n by Theorem 1.6. By Lemma 2.3, there exists a finite injectivemorphism C { x } −→ A and an injective morphism of maximal rank B −→ C { y } .The induced morphism C { x } −→ C { y } is strongly injective, since ϕ is stronglyinjective, and f is integral over C J x K . Thus we may assume that A = C { x } and B = C { y } .Let P ( x, z ) ∈ C J x K [ z ] be the minimal polynomial of f over C J x K . By replacing f by f − f (0) we may assume that f ∈ ( y ) C { y } . Consider the morphism ψ : C { x, z } −→ C { y } defined by ψ ( h ( x, z )) = h ( ϕ ( x ) , · · · , ϕ ( x n ) , f ) for any h ∈ C { x, z } . Then r F ( ψ ) = r F ( ϕ ) = n since P ( z ) ∈ Ker( b ψ ). Hence, because r F ( ψ ) > r( ψ ) > r( ϕ ), we have r( ψ ) = n . Thus, by ( I ), r A ( ψ ) = n so Ker( ψ ) is a height one primeideal, thus a principal ideal by [Mat89, Theorem 20.1]. Let Q ∈ C { x, z } be agenerator of Ker( ψ ). Then Q is a generator of Ker( b ψ ) and Q divides P in C J x, z K .Moreover P (0 , z ) = 0 since P ( x, z ) is a monic polynomial in z . Thus Q (0 , z ) = 0and, by the Weierstrass preparation Theorem, there exists a unit u ∈ C { x, z } suchthat uQ is a monic polynomial in z . But uQ ∈ Ker( ψ ), i.e. ( uQ )( ϕ ( x ) , f ) = 0, thus f is integral over C { x } .( II ) = ⇒ ( III ) First of all, we may assume that f is irreducible. Then we applythe result to each irreducible divisor of f . Let A = C { x } and B = C { x, t } / ( f ).We have f g = a ( x ) + a ( x ) t + · · · + a d ( x ) t d where the a i are in C J x K . There isa sequence of quadratic transforms τ = σ k ◦ · · · ◦ σ : C { x } −→ C { x } such that τ ( a d ( x )) = x α u ( x ) for some unit u ∈ C J x K . Thus τ ( f ) τ ( g ) = τ ( a ( x )) + τ ( a ( x )) t + · · · + x α u ( x ) t d . Therefore x ( d − α u − τ ( f ) τ ( g ) = y d + u − τ ( a d − ) y d − + u − x α τ ( a d − ) y d − + · · · + u − x ( d − α τ ( a )(22)where y := x α t . Let k ∈ C { x, t } be a prime divisor of τ ( f ). Then y ∈ C { x, t } / ( k )is integral over C J x K by (22). But the composed morphism C { x } / / C { x, t } / ( f ) τ / / C { x, t } ( k )has maximal rank, thus y is integral over C { x } by ( II ), and t is algebraic over C { x } .( III ) = ⇒ ( IV ) Let P ( x, t ) ∈ C J x K [ t ] be a nonzero polynomial in t such that P ( x, f ) = 0 mod. ( x − x z ). This means that there exists a formal power series g ∈ C J x, z K such that P ( x, f ( x, z )) + ( x − x z ) g ( x, z ) = 0 . We remark that, because f is algebraic over A , f + λz is algebraic over A for any λ ∈ C and, if deg t ( P ) = d , the polynomial T ( x, t ) := x d P ( x, t − λx /x ) ∈ C J x K [ t ] is a vanishing polynomial of f + λz . Let us choose λ ∈ C such that if h := f + λz then h (0 , z ) is a nonzero power series of order 1. We define I := ( t − h ( x, z ) , x − x z )as an ideal of C { x, t, z } . The ideal I is prime since C { x, t, z } /I ≃ C { x , . . . , x n , z } ,and ht( I ) = 2 since it is generated by two coprime elements.By the Weierstrass division Theorem t − h ( x, z ) = u ( x, z, t )( z + h ′ ( x, t ))for some unit u ( x, z, t ) ∈ C { x, z, t } and h ′ ( x, t ) ∈ C { x, t } . Thus I = ( z + h ′ ( x, t ) , x − x z ) . Since z + h ′ ( x, t ) and x − x z are coprime polynomials, I := I ∩ C { x, t } 6 = (0).Moreover I is a height one prime ideal so it is principal since C { x, t } is a uniquefactorization domain. Let Q ( x, t ) denote a generator of I , Q ( x, t ) = x + x h ′ ( x, t ).By the Weierstrass preparation Theorem we may assume that Q = x + k ( x , . . . , x n , t )for some k ∈ C { x , . . . , x n , t } . Moreover we can do the change of variables x x + k ( x , . . . , x n ,
0) and assume that t divides k ( x , . . . , x n , t ).On the other hand b I ∩ C J x, t K is also a height one prime ideal as for I , and b I ⊂ b I ∩ C J x, t K . But since I is prime, b I is also prime and b I = b I ∩ C J x, t K becauseboth have the same height. Hence Q is a generator of b I ∩ C J x, t K .Since T ( x, h ) = 0 modulo ( x − x z ), T ( x, t ) ∈ b I ∩ C J x, t K . Thus there is a formalpower series R ( x, t ) such that R ( x, t ) Q ( x, t ) = T ( x, t ) ∈ C J x K [ t ] . But (
III ) allows us to assume that R ( x, t ) ∈ C { x, t } and e T ( x, t ) := R ( x, t ) Q ( x, t ) ∈ C { x } [ t ]. Therefore, e T ( x, f + λx /x ) = 0, and f is algebraic over C { x } . This proves( IV ).( IV ) = ⇒ ( I ) We follow the beginning of the proof of Theorem 1.4: we argueby contradiction and assume that ϕ : C { x , x , x } −→ C { u , u } satisfies r( ϕ ) =r F ( ϕ ) = 2 and r A ( ϕ ) = 3. We will replace, step by step, the morphism ϕ byanother morphism ϕ ′ such that r( ϕ ′ ) = 2 and Ker( b ϕ ′ ) is generated by a Weierstrasspolynomial in x and such that ϕ ′ has a particularly simple form.First, we use Lemma 2.6 to assume that ϕ ( x ) = u and ϕ ( x ) = u α u β U ( u )where U ( u ) is a unit in C { u } . Now let σ ′ : C { u } −→ C { u } be defined by σ ′ ( u ) = u β and σ ′ ( u ) = u u α +12 . Then, we have σ ′ ◦ ϕ ( x ) = u β and σ ′ ◦ ϕ ( x ) = ( u u ) β ( α +1) V ( u )for some unit V ( u ). Therefore, up to ramification, we may assume that ϕ ( x ) = u and ϕ ( x ) = u u . Let P be a generator of Ker( b ϕ ). If we denote f ( u , u ) the image of x by ϕ , wehave P ( u , u u , f ( u , u )) = 0or f d ( u , u ) + a ( u , u u ) f d − ( u , u ) + · · · + a d ( u , u u ) = 0 ABRIELOV’S RANK THEOREM 57 or f d ( x , x ) + a ( x , x ) f d − ( x , x ) + · · · + a d ( x , x ) = 0 modulo ( x − x x ) . Therefore, by ( IV ), we may assume that the a i are in C { x , x } . But this impliesthat Ker( ϕ ) = (0) and r A ( ϕ ) < I ).6.3. Proof of Corollary 1.9.
As in the proof of ( I ) = ⇒ ( II ) in Theorem 1.7,we can assume that ϕ is injective, and ϕ : C { x } → C { y } . Since f is algebraic, thereis a nonzero polynomial P ( x, z ) ∈ C J x K [ z ] in the kernel of b ψ , where ψ is given by ψ : C { x, z } −→ C { y } h ( x, z ) h ( ϕ ( x ) , f )Once again as in the proof of ( I ) = ⇒ ( II ) in Theorem 1.7, Ker( ψ ) is a nonzeroprincipal ideal. Let Q ∈ Ker( ψ ). Since Q ∈ Ker( b ψ ), there is a formal power series g such that gQ = P ∈ C J x K [ z ]. Then, by Theorem 1.7( III ) there is h ∈ C { x, z } such that hQ ∈ C { x } [ z ]. Moreover hQ ∈ Ker( ψ ), which proves that f is integralover C { x } .6.4. Proof of Theorem 1.14. If f ( x d , . . . , x d n ) ∈ C { σ ∩ d Z n } is a root of P ( z ),then f ( x ) is a root of P ( x d , z ). Let Q ( z ) be the monic irreducible factor of P ( x d , z )in C J x K [ z ] having f ( x ) as a root. Since σ is strongly convex, there exists an invertiblelinear map L : Z n −→ Z n with positive coefficients such that L ( σ ) ⊂ R n > . Let( l i,j ) i,j be the matrix of L . Then, since L is the product of elementary matrices,the morphism τ : C { x } −→ C { x } defined by τ ( x i ) := x l ,i · · · x l n,i n for 1 i n is a composition of the map π defined by π ( x ) = x x , π ( x i ) = x i for i >
2, andof the maps permuting the variables x , . . . , x n . Since f ( x ) is a root of Q ( z ) then τ ( f ( x )) = f ( τ ( x )) is integral over τ ( C J x K ). Thus, by Theorem 1.7(II), f ( τ ( x )) isintegral over τ ( C { x } ) and the coefficients of Q ( z ) are convergent power series.Then the factors of P ( x d , z ) are the polynomials Q ( ξ x , . . . , ξ n x n , z ) for any d -th roots of the unity ξ , . . . , ξ n and they are in C { x } [ z ]. Thus the coefficients of P ( x d , z ) are convergent power series and the coefficients of P ( x, z ) also.7. Abhyankar-Jung Theorem
Abhyankar-Jung Theorem ([Ju08, Ab55]) . Let K be an algebraically closed fieldof characteristic zero. Let P ( y ) ∈ K J x K [ y ] be a monic polynomial in y of degree d such that ∆ P = x α u ( x ) where u (0) = 0 . Then the roots of P ( y ) are in K J x /d ! K . In the case of holomorphic polynomials, the Theorem admits a very simple proofbased on the local monodromy of solutions of the polynomial P (this is in fact theoriginal proof of Jung [Ju08], even if he stated the theorem only for the ring of con-vergent power series in 2 indeterminates). The formal case is much more involved,since the same geometrical arguments are unavailable. The first proof of the generalcase is due to Abhyankar [Ab55]. Kiyek and Vicente gave a modern proof of thisresult [KV04] and, recently, Parusiński and the third author have provided a moredirect proof reducing the general case to the complex case, via Lefschetz Principle[PR12]. In this section, we provide a new and very short proof of Abhyankar-JungTheorem, in all of its generality, following the techniques developed in § Proof of the Abhyankar-Jung Theorem.
We will prove this result in three steps:first the case where P ( y ) ∈ C { x } [ y ], then the case where P ( y ) ∈ C J x K [ y ], andfinally the general case. Step I:
Assume that P ( y ) ∈ C { x } [ y ]. Let ε > P ( y )are analytic on an open neighbourhood of the closure of D nε := { x ∈ C n | | x i | <ε, ∀ i } . The projection map ϕ : V := { ( x, y ) ∈ C n +1 | P ( x, y ) = 0 } −→ C n × { } is a branched covering, and its restriction over U := D nε \{ x · · · x n = 0 } is a finitecovering of degree d .Let a ∈ U . We have π ( U, a ) = Z n . Let W be a connected component of V ∩ ϕ − ( U ) and b ∈ ϕ − ( a ) ∩ W . The restriction of ϕ to W is a finite map ofdegree e d . Then the group extension ϕ ∗ ( π ( V ∩ ϕ − ( U ) , b )) ⊂ π ( U, a )is of finite index e . Therefore( e Z ) n ⊂ ϕ ∗ ( π ( V ∩ ϕ − ( U ) , b )) . This proves that the map ρ e defined by ρ e : z ∈ ( D ∗ ε /e ) n z e lifts to W , that is, there is an analytic map ψ : (cid:0) D ∗ ε /e (cid:1) n −→ W such that thefollowing diagram commutes: W ϕ (cid:15) (cid:15) (cid:0) D ∗ ε /d ! (cid:1) n ρ e / / ξ : : U .
Given a point c ∈ ( D ∗ ε /e ) n , such a ψ is uniquely defined by the choice of ψ ( c ).Since ϕ ( ψ ( c )) = ρ e ( c ), there is e such choices. Therefore there are e such distinctliftings. That is, there exist e analytic functions on ( D ∗ ε /d ! ) n , denoted by ξ , . . . , ξ e , such that ∀ i = 1 , . . . , e ∀ z ∈ ( D ∗ ε /e ) n , P ( z e , ξ i ( z )) = 0 . The polynomial P ( z e , y ) being monic, its roots ξ i ( z ) are bounded near the origin.Therefore, by Riemann Removable Singularity Theorem, they extend to analyticfunctions in a neighborhood of the origin.Now, we replace x by x e , and we replace P by Q that is defined by P ( x e , y ) = Q ei =1 ( y − ξ i ( x )) Q ( x, y ). This allows us to prove the result by induction on d , since( d − e )! e divides d ! (indeed e divides necessarily one of the integers ( d − e + 1) , . . . , d ). Step II:
Now suppose that P ( y ) ∈ C J x K [ y ], and ∆ P = x α u ( x ). We denote by K analgebraic closure of the field C (( x )). The valuation ord on C (( x )) extends on K , andthis extension is again denoted by ord. Denote by ξ , . . . , ξ d the roots of P ( y ) in K .If P ( y ) = y d + a ( x ) y d − + · · · + a d ( x ), there is a universal polynomial ∆ =∆( A , · · · , A d ) ∈ Q [ A , . . . , A d ], such that ∆ P = ∆( a , . . . , a d ).Now we apply Artin approximation Theorem: for every integer c , there are a c, ( x ), . . . , a c,d ( x ), u c ( x ) ∈ C { x } such that∆( a c, , . . . , a c,d ) = x α u c ( x ) ABRIELOV’S RANK THEOREM 59 and a i − a c,i , u − u c ∈ ( x ) c . In particular, for c >
1, we have u c (0) = 0. We set P c ( y ) = y d + P di =1 a c,i ( x ) y d − i . By Step I, there exist ξ c, , . . . , ξ c,d ∈ C { x /d ! } suchthat P c ( y ) = d Y i =1 ( y − ξ c,i ) . By Lemma 5.33, after renumbering the ξ c,i we may assume that ord( ξ c,i − ξ c ) goesto infinity. Therefore ξ c ∈ C J x /d ! K , and the result is proved. Step III:
Finally, in the general case, we denote by K the subfield of K generatedby all the coefficients of the series defining P ( y ). Such a field K can be embeddedin C , because Q −→ K is a field extension of finite or countable degree, while thedegree of Q −→ C is uncountable and C is algebraically closed. We denote by ι this embedding. Then, by the previous case, the roots of P ( y ) are in C J x /d ! K . Let K be the subfield of C generated by ι ( K ) and the coefficients of the roots of P ( y )in C J x /d ! K . Then, there is an embedding of K in K whose restriction to ι ( K ) is ι − , because the degree of ι ( K ) −→ K is 0 (by Lemma 7.1 given below), and K is algebraically closed. (cid:3) Lemma 7.1.
Let K ⊂ L be two fields. Let f ∈ L J x K be algebraic over K J x K . Let K be the field extension of K generated by the coefficients of f . Then K −→ K isan algebraic field extension.Proof. This result is well-known (see for instance [Gi69]). The proof goes as follows:we fix a monomial order on L J x K such that every subset of N n has a minimal element.For f := P α ∈ N n ∈ L J x K , we setexp( f ) := min { α ∈ N n | f α = 0 } and in( f ) := f exp( f ) x exp( f ) . Assume that P ( x, f ( x )) = 0 where P ( x, y ) ∈ K J x K [ y ] is nonzero. From this relationwe obtain that in( f ) is algebraic over K J x K , therefore f exp( f ) is integral over K .Then we replace f by f − in( f ) and we replace K by K ( f exp( f ) ). The result followsby induction. (cid:3) References [Ab55] S. Abhyankar, On the ramification of algebraic functions,
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