aa r X i v : . [ m a t h . A C ] F e b Minimal Pairs, Truncations and Diskoids
Andrei Benguş-Lasnier
Abstract
We build on the correspondence between abstract key polynomials and minimal pairs made by Novacoski andshow how to relate the valuations that are generated by each object. We can then give a geometric interpretationof valuations built in this fashion. To do so we employ an object called diskoid, which is a generalisation of theclassical concept of ball in non-archimedian valued fields.
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42 Truncation and Abstract Key Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63 Newton polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84 Minimal Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115 Correspondence between Minimal Pairs and Key Polynomials . . . . . . . . . . . . . . . . . . . . . . . .
146 Diskoids, towards a geometric interpretation of truncations . . . . . . . . . . . . . . . . . . . . . . . . . Introduction
Extending valuations from a field K onto a simple polynomial ring K [ X ] can be encoded in different ways andfollowing different strategies. MacLane first introduced key polynomials and augmented valuations in order tostudy simple extensions of discretely valued fields of rank [20]. This allowed him to prove results in classicalvaluation theory [21]. As the problem of local uniformization began to raise renewed interest, so did the studyof the extension problem of valuations from K to K [ X ] . Problems of ramifications and the study of defect ofextensions of valuations are deeply related to local uniformization as it has been shown in [10], [9].The problem became an inductive step in understanding the valuations of function fields of algebraic varieties.In dimension , the problem has a geometric flavour:– The theory of plane curves and jet schemes has been used to give a precise description of minimal generatingsequences of valuations in dimension 2 ([28], [2], [22], [13]).– Curvettes are being used in [27] whose results have later been generalised to non-algebraically closed fields byCutkosky and Vinh in [11].Our goal in this paper is to establish a correspondence between certain key polynomials and geometric objectsthat would give a faithful correspondence between the valuations induced by these key polynomials and the val-uations given by these geometric objects. We briefly mention the various notions that are involved in our studystarting with key polynomials. Different notions of key polynomials appeared since MacLane’s first definition in1936. Vaquié generalised this first approach in a series of articles ([31], [32], [30]) where he defines limit key poly-nomial and limit valuation in order to extend MacLane’s theory to non-necessarily discrete valuations of rank .Independently Spivakovsky, Olalla Acosta and Hererra Govantes gave a different approach to key polynomials, in[15],[14]. An alternative approach of key polynomials, which will be of interest to us, was introduced in [12], thatof abstract key polynomials (see Section 1). These allow to construct valuations, via a process called truncation:from µ a valuation on K [ X ] and a key polynomial Q , we consider for any f ∈ K [ X ] , its Q -expansion f = f + f Q + . . . + f n Q n , ∀ i, deg f i < deg Q and define a map µ Q , by setting 1 Q ( f ) = min i µ ( f i Q i ) . The fact that Q is an abstract key polynomial ensures that µ Q is a valuation.Our focus in this article revolves around the following conjecture. Conjecture.
Consider a valuation µ on K [ X ] and Q ∈ K [ X ] an abstract key polynomial for µ , such that µ ( Q ) ∈ Q ⊗ µ ( K × ) .1. There exists a subset ∆( Q ) ⊆ K that we shall call a diskoid , such that for every f ∈ K [ X ] , the minimum over ∆( Q ) , i.e., min x ∈ ∆( Q ) µ ( f ( x )) exists and is equal to the the truncation of µ along Q : ∀ f ∈ K [ X ] , µ Q ( f ) = min x ∈ ∆( Q ) µ ( f ( x )) . Furthermore, ∆( Q ) is a finite union of balls.2. There is a bijective correspondence between residually transcendental valuations µ over K [ X ] , which can begiven by a key polynomial Q , and diskoids ∆ = ∆( Q ) .What is remarkable here is that several key polynomials may induce the same truncated valuations, but theywould also induce the same diskoid ∆ , just like an open ball in a non-archimedian field may have all of its pointsas centres, yet induce the same valuation. We will prove the conjecture when K is henselian or µ | K is of rank one.The notion of diskoids mirrors in some sense the notion of maximal divisorial set for divisorial sets [16].A different approach to the extension problem came from a series of articles by Alexandru, Popescu andZăhărescu in the 90s [3], [4], [6], [5]. Their idea is to extend the problem to K , the algebraic closure of K .The problem is simpler to study over K [ X ] since all irreducible polynomials (among which, the key polynomials)are of degree . The authors coined the concept of minimal pairs in order to classify them. Consider a couple ( a, δ ) with a ∈ K and δ = µ ( X − a ) , where µ stands for an arbitrary extension of µ to K [ X ] . It will be called a minimalpair if deg( a ) is minimal among the deg( c ) , where c ∈ K such that µ ( X − c ) = δ . This allows to define a newvaluation on K [ X ] as follows µ a,δ ( f ) = min { µ ( a i ) + iδ } , f = a + a ( X − a ) + . . . + a n ( X − a ) n , a i ∈ K. We then can consider the trace of µ a,δ , by pulling it back down to K [ X ] ⊆ K [ X ] . Minimal pairs have used tostudy residually transcendental extensions and helped investigating invariants associated to elements of K . Thiswork is promoted especially by S. Khanduja (see for instance [18]).In this article we show that the valuations given by minimal pairs are in fact the same as the ones given byabstract key polynomials. Indeed we show that the trace of the valuation given by a minimal pair over K [ X ] ,is a truncated valuation of µ . We can already find a correspondence between minimal pairs and abstract keypolynomials in the work of Novacoski [23]. In order to get closer to our geometric interpretation we need to deepenthis correspondence. We do this in our Theorem 5.3. The correspondence can thus be stated as follows Theorem.
1. Consider Q the minimal polynomial of a , such that ( a, δ ) is a minimal pair, for some δ ∈ µ ( K [ X ] × ) .Then Q is an abstract key polynomial. Conversely, any abstract key polynomial Q has an optimising root a ,such that ( a, µ ( X − a )) is a minimal pair.2. Furthermore, one has an equality between valuations µ a,δ (cid:12)(cid:12) K [ X ] = µ Q . The first part of this theorem is the main goal of [23], the second part is our contribution to the theory. Similarresults have been stated and proven in [25, Theorem 5.1]. However they are made in the context of key polynomialsin the sense of MacLane and Vaquié whereas we work in the context of abstract key polynomials. Furthermore, theyprovide a proof only for augmented valuations of residually transcendental valuations and we do not need theseassumptions. One could imagine that we could prove our statement with the help of the work of Vaquié’s extensionof MacLane’s theory, however we would still need a similar result as [25, Theorem 5.1] but for limit-augmentedvaluations. Most of this work concerning these matters have been undertaken by M. Vaquié in [29]. Our proofwill make use of the structure of the graded algebras of a valuation which in our opinion, is an approach worth2oting. We will also give a couple of applications that we find noteworthy (see Proposition 5.8, Proposition 5.9 andTheorem 5.12).We also generalise [23, Proposition 3.1], by using a type of Newton polygon. This comes from our desire toencode the root configuration of a polynomial in the Newton polygon. The first slope of this polygon will be the δ invariant in the work surrounding minimal pairs.We sum up the contributions with their contributors in the following synoptic diagram (cid:26) Abstract keyPolynomials (cid:27) (cid:26)
MinimalPairs (cid:27)
Truncatedvaluationon K [ X ] Truncatedvaluationon K [ X ] (MacLane, Vaquié,Spivakovsky,...) (Alexandru,Popescu,Zăhărescu)NovacoskiextensionrestrictionOur work concerns the bottom part, the extension-restriction arrows.Our last section is concerned with establishing the geometric interpretation of residually transcendental valua-tions. To start off, valuations given by minimal pairs can be interpreted as minimal values of polynomials f ∈ K [ X ] attained on balls. Indeed take a minimal pair ( a, δ ) , so that we can consider the ball D ( a, δ ) = { b ∈ K | ν ( b − a ) > δ } . This type of set is not reduced to a single point if rk ( µ ) = rk ( ν ) . Indeed if for instance rk ( µ ) = rk ( ν ) + 1 , have δ be strictly in the last convex subgroup of µ ( K ( X ) ∗ ) , i.e., let δ be bigger than the value group of K . Then for any x ∈ K ∗ , δ > ν ( x ) . In this case D ( a, δ ) = { a } . In order to avoid such situations we will restrict our focus to residually transcendental valuations. In this contextrk ( µ ) = rk ( ν ) and thus D ( a, δ ) is non-reduced to a point, if δ < ∞ . Then one considers a valuation ν D ( a,δ ) definedas follows ν D ( a,δ ) ( f ) = min x ∈ D ( a,δ ) ν ( f ( x )) . We can prove that the minimum is attained and explicit. Indeed we show that ν D ( a,δ ) = ν a,δ . Furthermore this allows for a clear bijection between residually transcendental valuations and balls. Two minimalpairs ( a, δ ) , ( a ′ , δ ′ ) may yield the same valuation, however that is only the case if they yield the same ball D ( a, δ ) = D ( a ′ , δ ′ ) . We wish to build a similar bijection over K , by finding what we will call diskoids ∆( Q ) ⊆ K for keypolynomials Q . These will need to verify µ Q = ν ∆( Q ) , where ν ∆( Q ) ( f ) = min x ∈ ∆( Q ) ν ( f ( x )) ν ∆( Q ) = ν ∆( Q ) = ⇒ ∆( Q ) = ∆( Q ) . We can not just take ∆( Q ) to be balls, since different balls can induce the same valuation over K . If for instance µ Q is the same as ν D ( a,δ ) (cid:12)(cid:12) K [ X ] and σ ∈ Aut K ( K ) such that ν ◦ σ = ν , then µ Q is also the same as the restrictionto K [ X ] of ν σ ( D ( a,δ )) . We will use bigger subsets of K . We also want them to have a manageable shape and beeasy to grasp. We will give a reasonable candidate, called diskoids, that was studied by Julian Rüth in his PhD3hesis [26]. We will see how diskoids decompose in simple balls and how the absolute Galois group of K , i.e., G K = Gal ( K sep /K ) = Aut K ( K ) acts on these balls. We wish to show that a truncated valuation (by a fixedabstract key polynomial Q ) is the valuation given by a diskoid (associated to Q ) and that this correspondence is infact a bijection. This is what we wish to call our geometric interpretation. We prove our statement when ( K, ν ) ishenselian or of rank . Acknowledgements.
The author wishes to thank Steven Dale Cutkosky, Franz-Viktor Kuhlmann, HusseinMourtada, Bernard Teissier and the anonymous referee for the ample discussions, corrections, remarks, suggestionsand for their interest in this work.
In this paper we adopt the convention that the set N contains : N = { , , , , . . . } . We will write N ∗ for thepositive integers: N ∗ = { , , , . . . } . We start by defining our most basic objects, valuations. Let R be acommutative, unitary ring and Γ a totally ordered abelian group. We write + for the additive law of this groupand we add an element, not in Γ , that we denote ∞ . We write Γ ∞ = Γ ∪ {∞} and extend the addition and orderrelation so that ∞ plays the role of a biggest element. Definition 1.1. A valuation ν on R , is a map ν : R −→ Γ ∞ satisfying(V1) ∀ a, b ∈ R, ν ( a · b ) = ν ( a ) + ν ( b ) .(V2) ∀ a, b ∈ R, ν ( a + b ) > min { ν ( a ) , ν ( b ) } . This is called the ultrametric inequality.(V3) ν (1) = 0 and ν (0) = ∞ .(V4) ν − ( ∞ ) = (0) We often write ( R, ν ) for a valuative pair , i.e., a ring equipped with a valuation. Remark 1.2.
1. When we talk of embedding a pair ( R, ν ) into ( S, µ ) , we mean we set an injective morphism R ι ֒ −→ S , such that µ ◦ ι = ν .2. (V4) implies that R is a domain. One could then extend ν to a valuation over Q ( R ) , the quotient field of R ,by setting ν ( a/b ) = ν ( a ) − ν ( b ) . It is easy to see it is well-defined.Our study of valuations requires a powerful tool, the graded algebra of a valuation. For γ ∈ ν ( R \ { } ) anddefine the following groups P γ = P γ ( R, ν ) = { a ∈ R | ν ( a ) > γ }P + γ = P + γ ( R, ν ) = { a ∈ R | ν ( a ) > γ } . The graded ring gr ν ( R ) is gr ν ( R ) = M γ ∈ ν ( R \{ } ) P γ P + γ . This comes equipped with a map, called the initial form, in ν : R \ { } −→ gr ν ( R ) assigning to a its class modulo P + γ ( R ) with γ = ν ( a ) . By definition, any homogeneous element is thus represented and any initial form of anynon-zero element is non-zero. We can assign to the value in the graded ring. Remark 1.3.
For general filtered modules (or rings, algebras etc.) there is also a notion of initial form whichfails to be a morphism in general. It may not even be multiplicative, however when considering the graded algebraassociated to a valuation, we can still state some simple rules of computation:1. in ν ( a · b ) = in ν ( a ) · in ν ( b ) , ∀ a, b ∈ R . Thus the graded ring is an integral domain.2. if ν ( a ) > ν ( b ) then in ν ( a + b ) = in ν ( b ) .3. if ν ( a ) = ν ( b ) < ν ( a + b ) then in ν ( a + b ) = in ν ( a ) + in ν ( b ) = 0 .4. if ν ( a ) = ν ( b ) = ν ( a + b ) then in ν ( a + b ) = in ν ( a ) + in ν ( b ) .1. is simply a consequence of (V1) and the rest amount to using (V2).4 .2. Numerical Invariants. There are many ways in which one can measure the complexity of a valuation. Westart with assigning invariants to its value group.
Definition 1.4.
Consider a valuative pair ( R, ν ) and K the fraction field of R . The value group of ν written Φ( ν ) is Φ( ν ) := ν ( K × ) . We define the rank of ν , written rk ( ν ) rk ( ν ) := ord { convex subgroups of Φ( ν ) } where ord denotes the ordinal type of the set. We restrict ourselves to the cases where indeed, the set of convexsubgroups (also called isolated subgroups in the literature) do form a well-ordered set. Next we assign to ν its rational rank , written rat.rk ( ν ) rat.rk ( ν ) := dim Q Φ( ν ) ⊗ Z Q . Remark 1.5.
It is quite possible that ν ( R \ { } ) is a lot smaller than Φ( ν ) , however it is clear that ν ( R \ { } ) generates Φ( ν ) as a subgroup. Definition 1.6.
The valuation ring of the pair ( R, ν ) is a local subring of K , written O ( ν ) with maximal ideal m ( ν ) and residual field κ ( ν ) . We define them as follows: O ( ν ) = { a ∈ K | ν ( a ) > } m ( ν ) = { a ∈ K | ν ( a ) > } κ ( ν ) = O ( ν ) / m ( ν ) . By extension of valued rings, we mean an extension of rings R ⊆ S , each equipped with a respective valuation ν and µ , such that µ restricts to ν on R , µ (cid:12)(cid:12) R = ν. This is equivalent to saying that the natural inclusion map
R ֒ → S defines an embedding of valued pairs ( R, ν ) ֒ → ( S, µ ) . Given such an extension one can compare value groups and residual fields. Indeed, one has ν ( R \ { } ) ⊂ µ ( S \ { } ) , thus Φ( ν ) ⊂ Φ( µ ) . Secondly we have O ( ν ) ⊆ O ( µ ) and O ( ν ) ∩ m ( µ ) = m ( ν ) thus giving us a naturalinclusion of residual fields κ ( ν ) ⊆ κ ( µ ) . We will call them value group extension and residual extension respectively. Proposition 1.7.
We have the following inequalities given for any valuation ν on K and any valued extension ( L, µ ) :1. rk ( ν ) rat.rk ( ν ) .2. dim Q (Φ( µ ) / Φ( ν )) ⊗ Z Q + tr.deg κ ( ν ) κ ( µ ) tr.deg K L when these quantities are well-defined and finite. The second one is called the Zariski-Abhyankar inequality (see [7,Ch. 6, §10, no. 3, Cor. 1] for a proof). One could even generalise it (see [33, Appendix 2, Prop. 2] or [1, Theorem1]). This last result allows us to say that the rank of a valuation µ on K [ X ] , extending a valuation ν on K can notjump more than once rat.rk ( ν ) rat.rk ( µ ) rat.rk ( ν ) + 1 . We will say that our (simple transcendental) extension is valuation algebraic if the quotient group Φ( µ ) / Φ( ν ) istorsion and if the residual extension κ ( µ ) /κ ( ν ) is algebraic. Otherwise it is valuation transcendental . It will becalled value transcendental if Φ( µ ) / Φ( ν ) is not torsion and residually transcendental if κ ( µ ) /κ ( ν ) is not algebraic.The Abhyankar inequality tells us that either cases are possible, but not both at the same time.5 Truncation and Abstract Key Polynomials
Given two polynomials f, Q ∈ K [ X ] , deg Q > , one can define by successive euclidean divisions, the Q -expansionof f f = f + f Q + . . . + f n Q n , ∀ i, deg f i < deg Q. This expansion is unique and we write deg Q ( f ) the biggest i such that f i = 0 . When given a valuation µ on K [ X ] one can then define the truncated map of µ , with respect to Qµ Q ( f ) = min i µ ( f i Q i ) . For f ∈ K [ X ] and Q, µ as above, we will write S Q,µ ( f ) = { i ∈ N | µ ( f i Q i ) = µ Q ( f ) } or just S Q ( f ) for short, when the µ will be fixed once and for all. We also define d Q,µ ( f ) = d Q ( f ) := max S Q,µ ( f ) .The truncated map is not always a valuation. It still is a map extending the valuation ν = µ (cid:12)(cid:12) K and it stillverifies (V2) and (V3) (we assume that our valuation has trivial support and thus, µ Q also checks (V4)). It mayhowever fail to verify (V1). See [24, Example 2.5] for a counter-example. There is a natural condition for which itis a valuation, that is if Q is an abstract key polynomial. In that case we will talk about truncated valuation in Q .We will need to define the ǫ -level or ǫ factor of a polynomial f . Definition 2.1.
For any valuation µ on K [ X ] and polynomial f ∈ K [ X ] , just as above, we define ǫ µ ( f ) = max i > µ ( f ) − µ ( ∂ i f ) i . We will call it the ǫ factor of f . For our fixed µ we will simply write ǫ ( f ) = ǫ µ ( f ) and when we will truncate by Q ,we will write ǫ Q ( f ) = ǫ µ Q ( f ) .Here ∂ i represents the formal Hasse-Schmidt derivative on K [ X ] . This is an operator that can be defined bymeans of the Taylor expansion of polynomials in two variables f ( X + Y ) = X i > ∂ i f ( X ) Y i . By multiplying together the Taylor expansions of two polynomials f, g ∈ K [ X ] , we see that these Hasse-Schmidtderivatives satisfy the Leibniz rule ∂ i ( f · g ) = X i = j + k ∂ j f · ∂ k g. Furthermore we can compose Hasse-Schmidt derivatives. By expanding f ( X + Y + Z ) in two different ways, wecan show that ∂ i ◦ ∂ j = (cid:18) i + jj (cid:19) ∂ i + j . We now define abstract key polynomials.
Definition 2.2. [12, Definition 11] Let Q ∈ K [ X ] be a monic polynomial. We say that Q is an abstract keypolynomial for µ (we will abbreviate ABKP) if, for any f ∈ K [ X ]deg f < deg Q = ⇒ ǫ µ ( f ) < ǫ µ ( Q ) . As basic examples, any degree one polynomials are abstract key polynomials according to this definition. Lessobvious examples are the key polynomials given by MacLane-Vaquié’s key polynomials. See [12, Section 3].Now let us go through some basic properties the ABKPs verify.
Proposition 2.3. [24, Prop. 2.4 (ii)] Abstract key polynomials are irreducible.6 roposition 2.4. [12, Proposition 13] Let P , . . . , P t ∈ K [ X ] be polynomials of degree < deg Q (assume t > ). Ifwe set the following euclidean division Q ti =1 P i = qQ + r in K [ X ] , with deg r < deg Q , then µ t Y i =1 P i ! = µ ( r ) < µ Q ( qQ ) µ ( qQ ) . Remark that in the above proposition, r can not be , since Q is irreducible, thus Q ti =1 P i is prime with Q , sinceeach P i is prime with Q . Corollary 2.5. [12, Proposition 15] If Q is an ABKP, then µ Q is a valuation.This allows us to define the graded algebra of µ Q . We wish to understand its structure, but we first need thefollowing. Corollary 2.6. [12, Remark 16] Let α = deg Q and define G <α = X deg f< deg Q gr ν ( K ) · in µ Q ( f ) ⊆ gr µ Q ( K [ X ]) then this gr ν ( K ) -module is stable under multiplication and thus is an algebra. Remark 2.7.
We will see later (see Proposition 5.9) that the condition deg f < deg Q under the sum can bereplaced by ǫ µ ( f ) < ǫ ( Q ) .This subring G <α will play the role of coefficients in the following proposition. Proposition 2.8. [12, Remark 16] The graded ring of µ Q on K [ X ] has a simple polynomial structure. Moreprecisely gr µ Q ( K [ X ]) = G <α (cid:2) in µ Q Q (cid:3) with in µ Q Q transcendental over G <α .Let us just describe the action of the initial form here. Consider a polynomial f ∈ K [ X ] and its Q -expansion f = f + f Q + . . . + f n Q n , ∀ i, deg f i < deg Q so that in µ Q ( f ) = X j ∈ S Q ( f ) in µ Q ( f ) in µ Q ( Q ) j thanks to the rules of computation with initial forms. Thus in µ Q ( f ) ∈ G <α [ in µ Q ( Q )] and deg in µQ (cid:0) in µ Q ( f ) (cid:1) = max S Q,µ ( f ) . Truncation now gives us a valuation. Furthermore, we have that a truncation will bound the ǫ -level associatedto the truncation. We state the main result concerning this numerical properties, but we will postpone its proof(see Theorem 5.12) as we would have by then introduced the tools to prove it in a fairly straightforward way. Proposition 2.9. [12, Lemma 17] For any polynomial f ∈ K [ X ] ǫ µ Q ( f ) ǫ µ Q ( Q ) = ǫ µ ( Q ) . Furthermore we can even state the case of equality: ǫ µ Q ( f ) = ǫ µ Q ( Q ) ⇐⇒ S Q,µ ( f ) = { } . Newton polygons
In this section we will present a result concerning the values µ ( X − λ ) where λ runs through the roots of a polynomial f . We will call this data the root configuration of f . We will use Newton polygons. The first slope of the polygonwill be the ǫ factor. It is our hope that the whole slope data will lead us to a better understanding of the diskoidsdecomposition and the action of the absolute Galois group of K on them. Classical Newton polygons take a field F equipped an ultrametric absolutevalue | . | (or a valuation V of rank , so that one can suppose that V ( F × ) ⊆ R ). If however we want to work withfields of higher ranks, one needs to work out what convex sets are in R × Φ for any ordered abelian group Φ . Ourpresentation here takes many components from Vaquié’s presentation in [32]. We will consider the base changes Φ Q = Φ ⊗ Z Q and Φ R = Φ ⊗ Z R . Since Φ has no torsion, the canonical maps Φ Φ Q γ γ ⊗
1Φ Φ R γ γ ⊗ are all injective, thus we can consider Φ as a subgroup of Φ Q or Φ R .We define a line to be a subset L ⊆ R × Φ R defined by a linear or affine equation L = L q,α,β = { ( x, γ ) ∈ R × Φ R | qγ + αx + β = 0 } for some fixed values q ∈ R , α, β ∈ Φ R . The slope of L is given by s ( L ) = s ( L q,α,β ) = α/q whenever q = 0 . Thisdefinition of slope is not classical and corresponds to the negative of the natural slope of a line.There will always be a single line passing through two fixed and distinct points P , P , that we will denote ( P , P ) . Any line defines two half-spaces H L ≥ = { ( x, γ ) ∈ R × Φ R | qγ + αx + β ≥ } H L ≤ = { ( x, γ ) ∈ R × Φ R | qγ + αx + β ≤ } . For any subset A ⊆ R × Φ R , we define its convex hull byConv ( A ) = \ H half-space A ⊆ H H i.e., the intersection of half-spaces containing A . A set A is considered to be convex if Conv ( A ) = A . For any set A , we define its faces to be subsets F = Conv ( A ) ∩ L where L is a line in R × Φ R , satisfying- Conv ( A ) is contained in one of the half-spaces H L ≥ or H L ≤ .- F = Conv ( A ) ∩ L contains at least two points.We will say that L supports the face F . The slope s ( F ) of a face F of A will simply be s ( L ) where L supports F . Usually a Newton polygon is constructed for finite sets X = { ( k, γ k ) , k m } . We will write its Newtonpolygon as P N ( X ) = Conv ( { ( x, δ ) | ∃ ( x, γ ) ∈ X, δ ≥ γ } ) = Conv (( { } × Φ ≥ ) + X ) where Φ ≥ = { γ ∈ Φ R , γ ≥ } and A + B is the Minkowski sum of two subsets of R × Φ R . The bottom boundaryof P N ( X ) is then a finite polygonal line, thus describing it will be equivalent to giving the following- a finite sequence of non-negative integers a < a < . . . < a r = m (the abscissa or x -coordinates of thevertices of the polygonal line),- a finite set of values in Φ R : ǫ > . . . > ǫ r (the slopes of the segments forming the polygonal line)verifying 8igure 1: Example of a Newton polygon of a finite subset of R × R .1. ∀ k, t, k m, t r one has γ k + kǫ t > γ a t − + a t − ǫ t = γ a t + a t ǫ t .
2. if k < a t − or k > a t , then the above inequality is strict (this should be interpreted as ( a t , γ t ) being the pointswhere the polygonal line turns).The points A t = ( a t , γ a t ) are the vertices of the polygon and the faces are simply the segments [ A t − , A t ] , t r that have slope ǫ t . We will also call a t − a t − the length of this face.Let us now fix a valuation µ over K [ X ] with values in Φ . For any f ∈ K [ X ] define P N ( f, µ ) = P N ( X ( f, µ )) where X ( f, µ ) = { ( i, µ ( ∂ i f )) | i = 0 , . . . , deg( f ) } . Definition 3.1.
For any valuation µ on K [ X ] and polynomial f ∈ K [ X ] , we define the slope data f , as the sequenceof vertices of P N ( f, µ ) (cid:0) ( a , ǫ ) , . . . , ( a r , ǫ r ) (cid:1) as defined above.In the next subsection, we relate the finite data given by our version of the Newton polygon of f to someinformation given by the configuration of roots of f . In order to define what we mean by configuration of roots, we need to extend ourinitial valuation µ to K [ X ] where K is the algebraic closure of K . Decompose then f into linear factors f = α m Y i =1 ( X − λ i ) . What will be of interest to us will be the values µ ( X − λ i ) and how many times one such value is repeated. We canfix an indexation of the a i so that we have the following µ ( X − λ ) = . . . = µ ( X − λ l ) > µ ( X − λ l +1 ) = . . . = µ ( X − λ l + l ) ... > µ ( X − λ l + ... + l s − +1 ) = . . . = µ ( X − λ l + ... + l s ) . We will write µ ( X − λ l + ... + l t ) = δ t . Definition 3.2.
Set µ and f as above. Adopting the previous notations, we define the root configuration of f asthe finite sequence (cid:0) ( l , δ ) , . . . , ( l s , δ s ) (cid:1) . f , its root configuration and slope data are equivalent. Theorem 3.3.
Fix µ a valuation over K [ X ] and f ∈ K [ X ] . Then the root configuration of f, (cid:0) ( l t , δ t ) , t = 1 , . . . , s (cid:1) is encoded in its slope data (cid:0) ( a t , ǫ t ) , t = 1 , . . . , r (cid:1) as follows: s = ra t − a t − = l t , t = 1 , . . . , rǫ t = δ t , t = 1 , . . . , r where we have fixed a = 0 . Remark 3.4.
One can be surprised by the fact that even though the definition of the δ t needs us to choose anextension µ of µ to K , their value is ultimately independent of this choice.To prove Theorem 3.3 one can use a classic lemma concerning Newton polygons: Lemma 3.5.
Let ( F, V ) be any valued field and p ( T ) ∈ F [ T ] any polynomial, whose roots are in F . We write p ( T ) = c N Y k =1 ( T − c k ) = N X l =0 b l T l c, c k , b l ∈ F and consider the points { ( l, V ( b l )) , l = 0 , . . . , N } . If ζ is a slope of its Newton polygon of length ℓ , it follows thatprecisely ℓ of the roots c k have valuation ζ .A proof of Lemma 3.5 can be found in [17, Ch. IV, § 3] in the rank case, but it also applies for V of arbitraryrank as well. We can apply it to prove Theorem 3.3, by setting ( F, V ) = ( K ( X ) , µ ) and p ( T ) = f ( X + T ) = m X i =0 ∂ i f ( X ) T i ∈ K ( X )[ T ] . Indeed, the roots of p ( T ) , when considered as a polynomial of coefficients in K ( X ) are λ i − X, i = 1 , . . . , m . δ invariant. In [23], another quantity is defined in parallel to the ǫ factor. Definition 3.6.
For any valuation µ of K [ X ] and any polynomial f ∈ K [ X ] , we define δ as follows δ µ ( f ) := max { µ ( X − c ) | c root of f } . We will often abbreviate to δ ( f ) .Considering the root configuration (cid:0) ( l t , δ t ) , t = 1 , . . . , r (cid:1) of f , we obviously have by definition δ ( f ) = δ . On the other hand, if (cid:0) ( l t , δ t ) , t = 1 , . . . , r (cid:1) is the slope data of f , then we can write down ǫ ( f ) = ǫ . Thus [23, Prop. 3.1] is a consequence of our Theorem 3.3 for t = 1 . Remark 3.7.
We should observe that even though the choice µ is arbitrary, the δ µ ( f ) , when specifying f ∈ K [ X ] ,will only depend on µ . This short proof was suggested by Franz-Viktor Kuhlmann in a private communication. Minimal Pairs
In this section, we wish to build valuations over K [ X ] by bringing one down from K [ X ] . We start with ( K ( X ) , µ ) and extend the valuation to ( K ( X ) , µ ) . This can be represented in a simple diagram of extensions of valuative pairs ( K ( X ) , µ ) ( K ( X ) , µ )( K, ν ) (
K, ν ) Remark 4.1.
We can show that Aut K ( L ) ∼ = Aut K ( X ) ( L ( X )) for any algebraic field extension L/K . Indeed theisomorphism is given by simply taking a K ( X ) -automorphism σ of L ( X ) and restricting it to L . In order to seethat this mapping is well-defined, it is enough to show that for any ϕ ∈ Aut K ( X ) ( L ( X )) and any α ∈ L, ϕ ( α ) ∈ L .We know that α is algebraic over K , thus ϕ ( α ) is algebraic as well: there is a P ∈ K [ X ] such that P ( α ) = 0 . If ϕ ( α ) ∈ L ( X ) \ L , then ϕ ( α ) is transcendental over L and thus, transcendental over K , which is a contradiction.To see how to extend an element σ ∈ Aut K ( L ) to an element ˜ σ ∈ Aut K ( X ) ( L ( X )) , take any f = P k a k X k ∈ L [ X ] and set ˜ σ ( f ) = X k σ ( a k ) X k and this can then be extended to L ( X ) by setting ˜ σ ( f /g ) = ˜ σ ( a ) / ˜ σ ( b ) . This defines the reverse mapping to the onedefined above.From now on we will identify elements from the two groups. Lemma 4.2.
For any situation where we extend ν to ν over K and ν to µ over K [ X ] , there is a common extension µ , of both µ and ν . Proof.
We begin by taking any extension µ ′ of µ to K ( X ) . Since K/K is normal, Aut K ( K ) acts transitively onthe valuations of K extending ν , so that there is a σ ∈ Aut K ( X ) ( K ( X )) such that µ ′ ◦ σ (cid:12)(cid:12) K = ν .If we wish to construct a valuation on K [ X ] , we can consider a ∈ K , δ ∈ Φ , where Φ is a group containing Φ( ν ) ,and define ν a,δ X i a i ( X − a ) i ! = min i { ν ( a i ) + iδ } . We say that ( a, δ ) is a pair of definition for a valuation µ if µ = ν a,δ . Remark 4.3.
1. One observes that a i = ∂ i f ( a ) .2. If δ = µ ( X − a ) then ν a,δ is a truncation: ν a,δ = µ X − a .Several pairs of definition can yield the same valuation. We characterise these situations in the lemma below. Lemma 4.4.
Two pairs, ( a, δ ) and ( a ′ , δ ′ ) define the same valuation if and only if the following two conditions hold1. δ = δ ′ ν ( a − a ′ ) > δ There is a proof of this fact in the case of residually transcendental extensions in [3], but we give a proof in thegeneral case. 11 roof.
Consider two pairs ( a, δ ) , ( a ′ , δ ′ ) that define the same valuation. Then we have by definition δ ′ = ν a ′ ,δ ′ ( X − a ′ ) = ν a,δ ( X − a ′ ) = min { δ, ν ( a − a ′ ) } and by a symmetric argument we obtain δ = min { δ ′ , ν ( a − a ′ ) } thus δ ′ = δ and ν ( a − a ′ ) > δ .Conversely, consider pairs that verify the two conditions of the lemma and let us show that the valuations theydefine are equal. It is clear that they agree on K and it is sufficient to show they agree on polynomials of type X − b, b ∈ K to show that they are equal. Indeed, each polynomials factors as products of such simple degree 1polynomials, since K is algebraically closed. We have ν a,δ ( X − b ) = min { δ, ν ( a − b ) } ν a ′ ,δ ( X − b ) = min { δ ′ , ν ( a ′ − b ) } . If ν ( a − b ) > δ then ν a,δ ( X − b ) = δ , but ν ( a ′ − b ) = ν ( a ′ − a + a − b ) > min { ν ( a ′ − a ) , ν ( a − b ) } > δ so that ν a ′ ,δ ( X − b ) = δ = ν a,δ ( X − b ) . If ν ( a − b ) < δ then ν ( a ′ − b ) = ν ( a ′ − a + a − b ) = ν ( a − b ) thus ν a,δ ( X − b ) = ν ( a − b ) = ν ( a ′ − b ) = ν a ′ ,δ ( X − b ) . One can choose, among the { a ′ ∈ K | ν ( a ′ − a ) > δ } one of minimal degree. This leads us to the naturaldefinition of minimal pairs of definition. Definition 4.5.
We will say that ( a, δ ) is a minimal pair of definition for a valuation µ , if µ = ν a,δ , so that ( a, δ is a pair of definition for µ and deg K ( a ) is minimal among the pairs that define it.If we are given an arbitrary µ one can try to approximate it by setting δ = µ ( X − a ) so that ν ( a,δ ) µ . Thedefinition of minimal pair consists of a couple formed by an element a and the value δ = µ ( X − a ) . However, severalelements a may give the same δ . We impose furthermore that the degree of a over K is minimal. We have theprecise definition below. Definition 4.6.
We say that ( a, δ ) is a minimal pair for µ when1. µ ( X − a ) = δ .2. for any b ∈ K, deg K ( b ) < deg( a ) = ⇒ ν ( a − b ) < δ . Remark 4.7.
1. Condition 2 of Definition 4.6 above is also equivalent to the contrapositive 2’: b ∈ K, ν ( a − b ) > δ = ⇒ deg K ( b ) > deg( a ) .2. A minimal pair of definition characterises the valuation we are studying and allows for a direct way ofcomputing it. A minimal pair does not allow to compute the values of a polynomial: a valuation µ may havea minimal pair ( a, δ ) , however it may not be a minimal pair of definition for µ as ν a,δ may be different from µ .3. Minimal pairs (of definition) were defined in order to study residually transcendental valuation extensions,abbreviated RT extension ( i.e., the extension κ ( µ ) /κ ( ν ) is of transcendence degree ).RT extensions are characterised in the theorem below. Theorem 4.8. [3, Proposition 1, 2 and 3] The following are equivalent1. µ/ν is an RT extension.2. µ/ν is an RT extension.3. ∃ a ∈ K, δ ∈ Φ( ν ) such that µ = ν a,δ .4. Φ( µ ) = Φ( ν ) and µ ( X − K ) := { µ ( X − b ) , b ∈ K } has a maximal element.12urthermore, if one of these conditions is verified, we then have: max µ ( X − K ) = δ. Remark 4.9.
Item 3 of Theorem 4.8 will be refined for the extension ( K [ X ] , µ ) / ( K, ν ) once we have establishedTheorem 5.3 (see Proposition 5.8).When given a minimal pair ( a, δ ) , one can compute µ ( f ) for polynomials such that deg f < deg K ( a ) . This canbe stated more precisely in the lemma below. Lemma 4.10. [4, Theorem 2.1(a)] Fix ( a, δ ) a minimal pair. For any f ∈ K [ X ] with δ µ ( f ) < δ we havein µ X − a ( f ) = in µ X − a ( f ( a )) . Furthermore µ X − a ( f ) = µ ( f ) and µ X − a ( Q ) = µ ( Q ) where Q is the minimal polynomial of a over K .We need an auxiliary lemma, in order to prove the second point especially: Lemma 4.11.
For any f ∈ K [ X ] , and ( a, δ ) a minimal pair for µ , such that deg f < deg K ( a ) , then δ µ ( f ) < δ . Proof.
Consider any root of f, b ∈ K . Then deg K ( b ) deg f < deg K ( a ) , thus by the definition of minimalpairs, we have µ ( a − b ) < δ . Now suppose that b is such that δ µ ( f ) = µ ( X − b ) . By the ultrametric inequality µ ( X − b ) = µ ( X − a + ( a − b )) = µ ( a − b ) < δ .We give a presentation of the proof of Lemma 4.10 that differs from that of [4], as it shall serve the proof ofTheorem 5.3. Proof.
Consider the roots of f and write it as f = c d Y i =1 ( X − c i ) , with c, c i ∈ K. By hypothesis ∀ i, µ ( X − c i ) δ ( f ) < δ = δ ( Q ) = µ ( X − a ) . By the ultrametric property ν ( a − c i ) = µ (( X − c i ) − ( X − a ))= µ ( X − c i ) < µ ( X − a ) so we naturally have in µ X − a ( X − c i ) = in µ X − a ( X − a + a − c i ) = in µ X − a ( a − c i ) thus, by multiplicativity of in µ X − a in µ X − a ( f ) = in µ X − a ( c ) d Y i =1 in µ X − a ( X − c i )= in µ X − a ( c ) d Y i =1 in µ X − a ( a − c i )= in µ X − a c d Y i =1 a − c i ! = in µ X − a ( f ( a )) . It is an abstract key polynomial. Cf Theorem 5.1. µ X − a ( X − c i ) = min (cid:8) µ ( X − a ) , ν ( a − c i ) (cid:9) = ν ( a − c i )= µ ( X − c i ) . We now prove µ X − a ( Q ) = µ ( Q ) . Indeed first we have δ = ǫ µ ( Q ) = max i > µ ( Q ) − µ ( ∂ i Q ) i thus µ ( Q ) = min i > { µ ( ∂ i Q ) + iδ } . Furthermore / ∈ S X − a ( Q ) (because Q ( a ) = 0 so ν ( Q ( a )) = ∞ ), thus µ X − a ( Q ) = min i > (cid:8) ν ( ∂ i Q ( a )) + iδ (cid:9) . Thirdly, for any i > , deg ∂ i Q < deg Q = deg K ( a ) , so that δ µ ( ∂ i Q ) < δ by Lemma 4.11, so from the first part ofthe proof µ ( ∂ i Q ) = ν ( ∂ i Q ( a )) . Putting it all together µ ( Q ) = min i > { µ ( ∂ i Q ) + iδ } = min i > { ν ( ∂ i Q ( a )) + iδ } = µ X − a ( Q ) . In this section we wish to relate the extensions of valuations given by minimal pairs and those given by truncationby ABKPs. One such stride has been made in the work of Novacoski. We cite his result in the following theorem.
Theorem 5.1. [23, Proposition 3.2] Let a ∈ K be a root of an irreducible polynomial Q ∈ K [ X ] verifying δ = δ ( Q ) = µ ( X − a ) . Then Q is an ABKP for µ ⇐⇒ ( a, δ ) is a minimal pair for µ. Definition 5.2.
For any polynomial f ∈ K [ X ] we shall call a ∈ K an optimising root of f , if a is a root of f and µ ( X − a ) = δ ( f ) .We will prove that the truncated valuation µ Q , with Q an ABKP, comes as a restriction to K [ X ] of truncatedvaluations on K [ X ] , i.e., defined via a minimal pair. Theorem 5.3.
Let Q ∈ K [ X ] and let a ∈ K be an optimising root of Q . Then Q is an ABKP, if and only if ( a, µ ( X − a )) is a minimal pair. Furthermore µ X − a is an extension of µ Q µ X − a (cid:12)(cid:12) K [ X ] = µ Q thus inducing a natural injective map of graded algebras θ : gr µ Q ( K [ X ]) gr µ X − a ( K [ X ]) . It sends in µ Q ( f ) to in µ X − a ( f ) , for any f ∈ K [ X ] .This completes the correspondence between the situation over K and K . The proof is broken down into threesteps. 14 .1. Step 1. We first prove that the valuation given by the minimal pair is greater in value than the truncatedvaluation. This is a simple consequence of the ultrametric inequality.
Lemma 5.4. µ X − a (cid:12)(cid:12) K [ X ] > µ Q . Proof.
Fix f ∈ K [ X ] and set its Q -standard decomposition f = X i f i Q i thus by ultrametric inequality we have µ X − a ( f ) > min i µ X − a ( f i Q i ) . Now by Lemma 4.10 ∀ i, µ X − a ( f i ) = µ ( f i ) = µ ( f i ) and µ X − a ( Q ) = µ ( Q ) = µ ( Q ) . Thus yielding µ X − a ( f ) > min i µ X − a ( f i Q i ) = min i µ ( f i Q i ) = µ Q ( f ) . We build our morphism θ , based on the previous inequality. Corollary 5.5.
Lemma 5.4 induces a mapgr µ Q ( K [ X ]) gr µ X − a ( K [ X ]) θ Indeed, because of Lemma 5.4 one has the following inclusions P γ ( K [ X ] , µ Q ) ⊆ P γ ( K [ X ] , µ X − a ) P + γ ( K [ X ] , µ Q ) ⊆ P + γ ( K [ X ] , µ X − a ) for any γ ∈ Φ( µ Q ) ⊆ Φ( µ X − a ) . The map takes in µ Q ( f ) , which is an element in degree γ = µ Q ( f ) , and it sends it to f mod P + γ ( K [ X ] , µ X − a ) . Thus, if µ X − a ( f ) = µ Q ( f ) , then θ sends in µ Q ( f ) to in µ X − a ( f ) and if µ X − a ( f ) > µ Q ( f ) ,in µ Q ( f ) is sent to . Let us consider the kernel of θ . Considering the previous step, one hasKer ( θ ) = D in µ Q ( I ) E , where I = { f ∈ K [ X ] | µ X − a ( f ) > µ Q ( f ) } . Proving Theorem 5.3 amounts to showing that θ is injective. The main ingredient here is the structure of the gradedalgebra. According to Section 2 these have a polynomial presentationgr µ Q ( K [ X ]) = G <α [ T ] where T = in µ Q ( Q ) , α = deg Q and G <α = X deg f< deg Q gr ν ( K ) in µ Q ( f ) . We can make the same statement for µ X − a µ X − a ( K [ X ]) = ( gr ν ( K ))[ T ] where T = in µ X − a ( X − a ) . Indeed, by definition G < deg X − a is generated by initial forms of polynomials of degree < X − a ) , i.e., coefficients in K . This remark turns out to be crucial in many ways. One can use Lemma 4.10to prove the following proposition (which could be seen as a reformulation of Lemma 4.10). Proposition 5.6.
The map θ restricted to G <α induces an injective map between G <α and gr ν ( K ) ⊆ gr ν ( K )[ T ] .More explicitly, the θ map will take in µ Q ( f ) of deg( f ) < deg( Q ) and send it to in ν ( f ( a )) . We can now relateour different objects by displaying them in a commutative diagram G <α gr ν ( K ) gr µ Q ( K [ X ]) gr µ X − a ( K [ X ]) θθ Again by Lemma 4.10, we have that θ ( in µ Q Q ) = in µ X − a Q = 0 , because µ X − a ( Q ) = µ ( Q ) = µ Q ( Q ) . The initial formin µ X − a is multiplicative, so X − a dividing Q in K [ X ] implies that in µ X − a divides in µ X − a ( Q ) , so that d X − a,µ ( Q ) > .Hence in µ X − a ( Q ) / ∈ gr ν ( K ) . We will conclude by using the following basic result on polynomial rings.
Lemma 5.7.
Let φ : R −→ S be an injective integral domain map, that we extend to a map ˜ φ : R [ X ] −→ S [ Y ] which assigns to X a non-constant polynomial p ( Y ) ∈ S [ Y ] \ S . Then ˜ φ is again injective. Proof.
Consider a non-zero polynomial q ( X ) ∈ R [ X ] . If q ∈ R then by assumption ˜ φ ( q ) = φ ( q ) = 0 . Otherwisesuppose deg q ( X ) > . Since all the rings we are considering are integral domains, we get deg ˜ φ ( q ( X )) = deg q deg p > ⇒ ˜ φ ( Q ( X )) = 0 .Thus θ is injective. Furthermore, taking into account its construction, θ is homogeneous. This concludes ourproof. We start by refining Theorem 4.8.
Proposition 5.8. [23, Theorem 1.3] Our extension ( K [ X ] , µ ) / ( K, ν ) is residually transcendental if and only if thereis a polynomial Q such that µ = µ Q and µ ( Q ) is torsion over Φ( ν ) . Furthermore Q can be taken to be an ABKP. Proof.
Choose an extension µ of µ to K [ X ] , which restricts to ν over K ( ν is thus an extension of ν ). By Theorem 4.8,our extension is residually transcendental if and only if we are given a pair of definition ( a, δ ) ∈ K × Φ ν for µµ X − a ( X i a i ( X − a ) i ) = min i { ν ( a i ) + iδ } , where a i ∈ K. We can suppose, that this pair is minimal, simply by choosing a of minimal degree over K , among the pairsof definition of µ . Thus the minimal polynomial of Q of a is an ABKP and by our Theorem 5.3, we have that ( K [ X ] , µ ) / ( K, ν ) is residually transcendental if and only if µ = µ (cid:12)(cid:12) K [ X ] = ν a,δ (cid:12)(cid:12) K [ X ] = µ X − a (cid:12)(cid:12) K [ X ] = µ Q . Now recall Remark 2.7. We can now extend the conditionunder the sum, defining G <α in the proposition below. Proposition 5.9.
Set f ∈ K [ X ] , ǫ ( f ) < ǫ ( Q ) where Q is an ABKP for a valuation µ over K [ X ] . There exists g ∈ K [ X ] such that deg g < deg Q and in µ ( f ) = in µ ( g ) . Thus one can re-write the definition of G <α where α = deg Q : G <α = X ǫ ( f ) <ǫ ( Q ) gr ν ( K ) in µ Q ( f ) . i.e., µ ( Q ) ∈ Q ⊗ Φ( ν ) roof. We already have G <α ⊆ X ǫ ( f ) <ǫ ( Q ) gr ν ( K ) in µ Q ( f ) since deg f < deg Q implies ǫ ( f ) < ǫ ( Q ) by definition of ABKPs. Consider f with ǫ ( f ) < ǫ ( Q ) and its Q -expansion f = X i f i Q i , deg f i < deg Q. Consider an optimising root of Q , a ∈ K , thus providing us with a minimal pair ( a, δ ) , δ = δ µ ( Q ) = ǫ µ ( Q ) . Wethus have the following equalities θ (cid:0) in µ Q ( f ) (cid:1) = in µ X − a ( f ) ( ∗ ) = in µ X − a ( f ( a ))= in µ X − a ( f ( a )) ( ∗∗ ) = in µ X − a ( f )= θ (cid:0) in µ Q ( f ) (cid:1) . ( ∗ ) is a direct application of Lemma 4.10 and so is ( ∗∗ ) , since having deg f < deg Q , implies ǫ ( f ) < ǫ ( Q ) . Thus,since θ is injective, in µ Q ( f ) = in µ Q ( f ) ∈ G <α . Remark 5.10.
This result should then be compared to [20, Theorem 5.2].Informally, the polynomials that verify ǫ ( f ) < ǫ ( Q ) have their values known by the truncation (or equivalentlyby the corresponding minimal pair), thus their value is relatively easy to define or compute. We now prove the announced result Proposition 2.9, that for any f ∈ K [ X ] , ǫ µ Q ( f ) ǫ µ Q ( Q ) = ǫ µ ( Q ) . From the proof of Proposition 5.9, one can extract the fact that S Q ( f ) = { } whenever ǫ ( f ) < ǫ ( Q ) .This is proven in [12, Prop. 17, Prop. 18] and the authors of the article also announce the proof of its converse inan upcoming article. We now give a proof of these facts and also prove the converse of [12, Prop. 18]. Proposition 5.11.
For any polynomial f ∈ K [ X ] and any ABKP for a valuation µ over K [ X ] we have ǫ µ Q ( f ) ǫ µ Q ( Q ) = ǫ µ ( Q ) . Proof.
We will fix µ an extension of µ to K [ X ] and set a an optimising root of Q . By Theorem 5.3 we know that µ X − a is an extension of µ Q and by Theorem 3.3 we can write ǫ µ Q ( f ) = δ µ X − a ( f ) = max { µ X − a ( X − c ); c root of f } and ǫ µ ( Q ) = ǫ µ Q ( Q ) = µ ( X − a ) . Thus we simply need to show that for any c ∈ K µ X − a ( X − c ) µ ( X − a ) ,which is straightforward µ X − a ( X − c ) = min { µ ( X − a ) , ν ( a − c ) } µ ( X − a ) . Theorem 5.12. ∀ f ∈ K [ X ] , ǫ µ Q ( f ) ǫ µ Q ( Q ) = ǫ µ ( Q ) and we have equality if and only if there is a b > suchthat µ Q ( f ) − µ Q ( ∂ b f ) b = ǫ µ ( Q ) . In other words S Q,µ ( f ) = { } ⇐⇒ ǫ µ Q ( f ) = ǫ µ ( Q ) . Proof.
Let us henceforth write ǫ µ ( Q ) = ǫ . We choose an optimising root of Q , a ∈ K ( i.e., µ ( X − a ) = δ µ ( Q ) ), sothat one can apply Theorem 5.3. Since µ X − a is an extension of µ Q , one can write ǫ µ Q ( f ) = ǫ µ X − a ( f ) . Let us abbreviate the following initial forms 17 = in µ Q ( Q ) T = in µ X − a ( X − a ) . We can now express the following equivalences S Q,µ ( f ) = { } ⇐⇒ d Q ( f ) = deg T in µ Q ( f ) > ⇐⇒ d X − a ( f ) = deg T in µ Q ( f ) > ⇐⇒ S X − a,µ ( f ) = { } . Indeed, recall the properties of the map θ in the proof of Theorem 5.3. It sends initial forms of polynomials f with deg T in µ Q = 0 to an initial form with deg T in µ X − a f = 0 and initial forms of polynomials f with deg T in µ Q > , toinitial forms with deg T in µ X − a f > .Thus one can simply work in K [ X ] . Let f ∈ K [ X ] such that S X − a,µ = { } so d = d X − a ( f ) > µ X − a ( f ) = µ ( ∂ d f ( a )( X − a ) d ) . But since we have, by definition of truncation, the inequality µ ( ∂ d f ( a )) > µ X − a ( ∂ d f ) which implies µ X − a ( f ) = µ ( ∂ d f ( a )) + dǫ > µ X − a ( ∂ d f ) + dǫ thus ǫ = ǫ µ Q ( Q ) ( ∗ ) > ǫ µ Q ( f ) = ǫ µ X − a ( f ) > µ X − a ( f ) − µ X − a ( ∂ d f ) d > ǫ so everything is an equality (the inequality ( ∗ ) is just due to Proposition 5.11). Alternatively, one can start with ǫ Q ( f ) = ǫ µ X − a ( f ) < ǫ , so that ∀ i > , µ X − a ( f ) < µ X − a ( ∂ i f ) + iǫ ν ( ∂ i f ( a )) + iǫ thus implying that S Q ( f ) = { } .Let us now show the converse. We suppose, ad absurdum, that we have a polynomial f ∈ K [ X ] verifying thetwo following statements: ǫ X − a ( f ) = ǫ and S X − a ( f ) = { } . The first statement means that µ X − a ( f ) = µ X − a ( ∂ d f ) + dǫ (1)with d > . We choose d maximal with this property (so that d = deg T in µ X − a ( f ) > ). The second statementassures us that ∀ i > , µ X − a ( f ) = ν ( f ( a )) < ν ( ∂ i f ( a )) + iǫ. Putting these two together gives us µ X − a ( ∂ d f ) + dǫ = µ X − a ( f ) < ν ( ∂ d f ( a )) + dǫ or, by simplifying µ X − a ( ∂ d f ) < ν ( ∂ d f ( a )) . (2)This implies that S X − a ( ∂ d f ) = { } . Indeed, S X − a ( ∂ d f ) = { } means that µ X − a ( ∂ d f ) = ν ( ∂ d ( f ( a )) , contradicting (2) . By the first direction of the statement, proven above, we have that ǫ X − a ( ∂ d f ) = ǫ , meaning that we have a b > such that ǫ = µ X − a ( ∂ d f ) − µ X − a ( ∂ b ∂ d f ) b .e., µ X − a ( ∂ d f ) = µ X − a ( ∂ b ∂ d f ) + bǫ. We know that ∂ b ∂ d = (cid:0) b + db (cid:1) ∂ b + d . We now set p = char κ ( ν ) . If p = 0 then ν (cid:16)(cid:0) b + db (cid:1)(cid:17) = 0 , no matter the values of b, d . However, if p > , then ν (cid:18)(cid:18) b + db (cid:19)(cid:19) = ∞ or , whether p divides (cid:18) b + db (cid:19) or not . However we know in our situations that our quantities are finite, so we avoid the ∞ case, thus µ X − a ( ∂ d f ) = µ X − a ( ∂ b + d f ) + bǫ and combining with (1) we get µ X − a ( f ) = µ X − a ( ∂ b + d f ) + ( b + d ) ǫ which contradicts the maximality of d . In this last section, we will try and discuss a possible way of giving a geometric interpretation for residuallytranscendental valuations.
We start by studying the simplest possible construction: balls. We shall fix a valued field ( K, ν ) andan extension to its algebraic closure ( K, ν ) . Definition 6.1.
Fix a ∈ K and δ ∈ Φ( ν ) ∪ {∞} . Then we set D ( a, δ ) = { x ∈ K | ν ( x − a ) > δ } to be the closed ball centred around a and of radius δ . Lemma 6.2.
Let D = D ( a, δ ) closed ball. Then for any polynomial f ∈ K [ X ] , the following minimum is welldefined and can even be explicitly written min x ∈ D ν ( f ( x )) = min i ∈ N { ν ( ∂ i f ( a )) + iδ } . Thus the map f min x ∈ D ν ( f ( x )) , that we denote by ν D , is a valuation, which coincides with the valuation ν a,δ given by a (not necessarily minimal) defining pair ( a, δ ) . Proof.
Write f as f ( X ) = n X i =0 a i (cid:18) X − ab (cid:19) i , a i ∈ K where b ∈ K is an element such that ν ( b ) = δ . Since x ∈ D if and only if ν (cid:0) x − ab (cid:1) > , for such a x we have by theultrametric property ν ( f ( x )) > min i n ν ( a i ) . Let us now show that in fact this last value is assumed by f inside D . We first normalise the expression of f above, i.e., we divide by the a i of minimal value and we can assume min i n ν ( a i ) = 0 . Reducing the polynomial’scoefficients to the residue field κ ( ν ) we get a non-zero polynomial f ∈ κ ( ν )[ X ] . κ ( ν ) being algebraically closed, wecan find a value y ∈ κ ( ν ) such that f ( y ) = 1 . We can choose x ∈ K , such that x − ab is in O ( ν ) and reduces to y in κ ( ν ) , thus ν ( f ( x )) = 0 . Necessarily x ∈ D , since ν (cid:0) x − ab (cid:1) > . Remark 6.3.
1. According to Theorem 4.8, the valuation ν D is residually transcendental.19. If δ = ∞ , then D = D ( a, δ ) = { a } so we have min x ∈ D ν ( f ( x ))) = ν ( f ( a )) . We can restate and extend Theorem 4.8 in the following way.
Theorem 6.4.
There is a one-to-one correspondence between residually transcendental extensions and balls as wehave just defined. More precisely, we can define the following maps, which form a commutative diagram (cid:26)
Minimal Pairs ( a, δ ) with δ ∈ Φ( ν ) (cid:27) { RT Extensions } { balls } ( a, δ ) ν a,δ = ν D D = D ( a, δ ) ontoonto one-to-one Proof.
We already know that the above maps are well-defined. We also know that they are onto. Indeed, on onehand, any residually transcendental extension is given by a minimal pair as we have established in Theorem 4.8.On the other hand, by the ultrametric property we know that any point inside a ball D ( b, δ ) is a centre of the ball,thus we can choose one of minimal degree, which yields a minimal pair.Finally the horizontal map is injective. Indeed consider two balls D = D ( a, δ ) , D ′ ( a ′ , δ ′ ) , yielding the samevaluations ν D = ν D ′ . We can furthermore suppose that ( a, δ ) , ( a ′ , δ ′ ) are minimal pairs, thus we have ν a,δ = ν a ′ ,δ ′ and by Lemma 4.4, we have δ = δ ′ and ν ( a − a ′ ) > δ , which in turn is equivalent to D = D ′ . Remark 6.5.
If rk ( µ ) = rk ( ν ) + 1 , then there are elements δ ∈ Φ( µ ) \ Φ( ν ) , δ > , so that ∀ γ ∈ Φ( ν ) , δ > γ . Thus D ( a, δ ) = { a } . We thus see that there is no hope for a 1-1 correspondence between valuations and balls. Indeed ( a, δ ) and ( a, ∞ ) give the same balls, yet their valuations are different: ν ( a,δ ) ( X − a ) = δ < ∞ = ν ( a, ∞ ) ( X − a ) . The rational rank can increase without the rank increasing, but that would not affect the correspondence. Considerfor instance K = C (( t )) equipped with the t -multiplicity valuation: ν = ord t . The algebraic closure of K is thefield of Puiseux series with complex coefficients: K = S n ∈ N ∗ C (( t /n )) . We can extend ν to K , by extending the t -multiplicity. Over K [ X ] , we can extend ν , by considering the monomial valuation, associating to X the value √ .The resulting valuation µ is still of rank , its rational rank is , yet there is no problem as the one shown above: ∀ δ ∈ Φ( µ ) = Q ( √ , ∃ γ ∈ Φ( ν ) = Q , γ > δ . However, the valuation will not be RT. This valuation µ will be out ofour correspondence. In the future we could extend our correspondence to include such cases as well. Our wish is to generalise this statement up to truncated valuations µ Q which are residually tran-scendental ( i.e., such that µ ( Q ) ∈ Q ⊗ Φ( ν ) = Φ( ν ) ). More specifically we wish to find subsets D ⊆ K that canverify the following:(D1) ∀ f ∈ K [ X ] , ν D := min x ∈ D ν (cid:0) f ( x ) (cid:1) is well-defined.(D2) ν D is a valuation.(D3) There is a way of associating to Q , ABKP for µ with µ ( Q ) ∈ Q ⊗ Φ( ν ) , a subset D Q such that ν D Q = µ Q .(D4) The mapping D ν D between such subsets and RT extensions over K [ X ] is bijective.Our candidate was already used by Rüth in his PhD thesis [26] called diskoid. We wish for it to replace the roleof balls in the above correspondence. 20 efinition 6.6. For any polynomial f ∈ K [ X ] and any value ρ ∈ Φ( ν ) ∪ {∞} we define the diskoid centred at f ofradius ρ e D ( f, ρ ) = { x ∈ K | ν ( f ( x )) > ρ } . Observe that we then have e D ( X − a, δ ) = D ( a, δ ) . Diskoids are a generalisation of balls. Just like in the previoussection, we wish to show that for an abstract key polynomial Q of value µ ( Q ) ∈ Φ( ν ) , the following map f ∈ K [ X ] min x ∈ e D ( Q,µ ( Q )) ν ( f ( x )) is well-defined and corresponds to µ Q . This is not an easy task, but we can start by investigating the structure ofthese diskoids. We will show that in fact they are disjoint unions of balls. Remark 6.7.
If rk ( µ ) = rk ( ν ) + 1 , i.e., when there is an added convex subgroup when passing from ( K, ν ) to ( K [ X ] , µ ) , we have key polynomials Q such that µ ( Q ) ∈ Φ( µ ) \ Φ( ν ) , µ ( Q ) > , so that ∀ γ ∈ Φ( ν ) , γ < µ ( Q ) . Inthis case diskoids are reduced to finite sets of points: for any x ∈ K, ν ( f ( x )) ∈ Φ( ν ) , so ν ( f ( x )) < µ ( Q ) , unless x is a root of f , in which case ν ( f ( x )) = ∞ . Thus there is no hope for a 1-1 correspondence. For instance, if weconsider only valuations µ of rank , such problem will not arise. Indeed, in that case Φ( µ ) can be embedded as anordered group inside R .If µ is residually transcendental, such problems do not arise either, since in this case Φ( µ ) = Φ( ν ) and rk ( µ ) = rk ( ν ) .We need the following lemma. Lemma 6.8.
Set f ∈ K [ X ] , a ∈ K a root of f and ρ ∈ Φ( ν ) . Then the quantity ǫ ( a ; f, ρ ) = min { λ ∈ Φ( ν ) | D ( a, λ ) ⊆ e D ( f, ρ ) } is well-defined and can even be explicitly written ǫ ( a ; f, ρ ) = max i ∈ N ∗ ρ − ν ( ∂ i f ( a )) i . Proof.
Let us write f as f ( X ) = n X i =1 a i ( X − a ) i , a i ∈ K so that a i = ∂ i f ( a ) . We remark that a = 0 since f ( a ) = 0 , so considering Lemma 6.2, for any λ ∈ Φ( ν ) , min x ∈ D ( a,λ ) ν ( f ( x )) = min i > { ν ( a i ) + iλ } . Thus we can write the following equivalences D ( a, λ ) ⊆ e D ( f, ρ ) ⇐⇒ ∀ x ∈ D ( a, λ ) , ν ( f ( x )) > ρ ⇐⇒ min x ∈ D ( a,λ ) ν ( f ( x )) > ρ ⇐⇒ min i n { ν ( a i ) + iλ } > ρ ⇐⇒ λ > max i n ρ − ν ( a i ) i . Thus we can set the following expression ǫ ( a ; f, ρ ) = max i n ρ − ν ( a i ) i . We can now show how diskoids decompose.
Lemma 6.9.
For any f ∈ K [ X ] , ρ ∈ Φ( ν ) , e D ( f, ρ ) is a finite union of balls. They are centred around the roots of f : e D ( f, ρ ) = [ f ( c )=0 D ( a, ǫ ( c ; f, ρ )) . roof. Let c , . . . , c n be the possibly repeated roots of f in K . By Lemma 6.8 we can assign to each c i thevalue ǫ i = ǫ ( c i ; f, ρ ) ∈ Φ( ν ) such that D ( c i , ǫ i ) ⊆ e D ( f, ρ ) , with ǫ i minimal for this property. We now show that e D ( f, ρ ) ⊆ ∪ i D ( c i , ǫ i ) , the other inclusion being clear. Let x ∈ e D ( f, ρ ) n X i =1 ν ( x − c i ) = ν ( f ( x )) > ρ. We can rearrange the c i so that ν ( x − c ) > . . . > ν ( x − c n ) . In other words we have relabelled the roots to make c the closest root to x . Now we will show that D ( c , ν ( x − c )) ⊆ D ( f, ρ ) so that ǫ i ν ( x − c ) . For y ∈ D ( c , ν ( x − c )) we have ν ( f ( y )) = n X i =1 ν ( y − c i ) > ν ( x − c ) + n X i =2 ν ( y − c | {z } > ν ( c − x ) + c − x | {z } > ν ( x − c i ) + x − c i ) > ν ( x − c ) + n X i =2 ν ( x − c i ) > ρ. Remark 6.10. If ( a, δ ) is a minimal pair associated to an abstract key polynomial Q , then ǫ ( a ; Q, µ ( Q )) = δ. Indeed, since for any i > , deg ∂ i Q < deg Q , we get by Theorem 5.12 µ ( ∂ i Q ) = µ Q ( ∂ i Q ) = ν a,δ ( ∂ i Q ) = ν ( ∂ i Q ( a )) ,since ǫ µ Q ( ∂ i Q ) = ǫ µ ( ∂ i Q ) < ǫ µ ( Q ) . Definition 6.11.
For any polynomial f, g ∈ K [ X ] and value ρ ∈ Φ( ν ) , the following value min x ∈ e D ( f,ρ ) ν ( g ( x )) is well-defined, according to Lemma 6.8 and Lemma 6.2 (Diskoids satisfy (D1)). We can thus define the followingmap ν e D ( f,ρ ) : K [ X ] −→ Φ( ν ) ∪ {∞} g min x ∈ e D ( f,ρ ) ν ( g ( x )) . Proposition 6.12.
The map ν e D ( f,ρ ) is ultrametric (verifies (V2)). Proof.
Take g, h ∈ K [ X ] and set x ∈ e D ( f, ρ ) such that ν e D ( f,ρ ) ( g + h ) = ν ( g ( x ) + h ( x )) . We have the following ν e D ( f,ρ ) ( g + h ) = ν ( g ( x ) + h ( x )) > min { ν ( g ( x )) , ν ( h ( x )) } > min { ν e D ( f,ρ ) ( g ) , ν e D ( f,ρ ) ( h ) } . emark 6.13. In a similar way one can prove that ∀ g, h ∈ K [ X ] , ν e D ( f,ρ ) ( gh ) > ν e D ( f,ρ ) ( g ) + ν e D ( f,ρ ) ( h ) . However ν e D ( f,ρ ) may fail to be multiplicative (the condition (V1) for valuations) so diskoids fail to satisfy (D2). Wegive a simple counter-example: set a ∈ K such that ν ( a ) < . Define D a = D ( a, , D = D (0 , and D = D a ∪ D .It can be realised as a diskoid of f = X ( X − a ) e D ( X ( X − a ) , ν ( a )) = D ( a, ∪ D (0 ,
0) = D since ǫ (0; f, ν ( a )) = ǫ ( a ; f, ν ( a )) = max (cid:26) ν ( a )2 , ν ( a ) − ν ( a ) (cid:27) = 0 . Indeed ∂ X ( X − a ) = 2 X − a and ∂ X ( X − a ) = 1 . Then we have ν D ( X ) = 0 ν D a ( X ) = ν ( a ) ν D ( X − a ) = ν ( a ) ν D a ( X − a ) = 0 so clearly ν e D ( f,ν ( a )) ( X ) = ν e D ( f,ν ( a )) ( X − a ) = ν ( a ) . Furthermore ν e D ( f,ν ( a )) ( X ( X − a )) = ν ( a ) but ν e D ( f,ν ( a )) ( X ) + ν e D ( f,ν ( a )) ( X − a ) = 2 ν ( a ) < ν ( a ) = ν e D ( f,ν ( a )) ( X ( X − a )) hence, ν e D ( f,ν ( a )) is not a valuation.One can hope that if we restrict to studying only abstract key polynomials Q , with µ ( Q ) ∈ Φ( ν ) , the diskoids e D ( Q, µ ( Q )) yield true valuations. One could even generalise Theorem 6.4 and interpret the decomposition of adiskoid into balls at the level of valuations. Let us state these as conjectures. Conjecture 6.14. (C1) If Q is an ABKP for µ with µ ( Q ) ∈ Φ( ν ) , then ν e D ( Q,µ ( Q )) is a valuation.(C2) If µ is an RT extension, there is a unique diskoid D such that µ = ν D .In order to explore these questions, we will recall some basic facts about henselization. In this section we recall some facts about theinteraction between extensions of valued fields and the Galois theory of the corresponding field extension. We startwith the following crucial notion.
Definition 6.15.
A valued field ( K, ν ) is called henselian if there is only one extension of ν to the algebraic closureof K .Several ways to characterise henselian fields exist and relate more or less with Hensel’s lemma (one can consult[19] for a deeper understanding). Example 6.16.
Henselian fields include discrete valued fields of rank which are complete, such as the field of p -adic numbers Q p equipped with its natural p -adic absolute value, or the formal power series k (( t )) where k is anyfield, which we equip with the natural t -adic valuation.Not every field is henselian, but one can find a smallest henselian extension of a valued pair ( K, ν ) which ishenselian. It is called a henselization of ( K, ν ) . Definition 6.17.
An extension ( ˜ K, ˜ ν ) of ( K, ν ) is called a henselization of ( K, ν ) if it is henselian and if forevery henselian valued field ( E, ζ ) and every embedding λ : ( K, ν ) ֒ → ( E, ζ ) there exists a unique embedding ˜ λ : ( ˜ K, ˜ ν ) ֒ → ( E, ζ ) extending λ . 23enselizations exist and can be constructed in the following way. Choose a separable closure K sep of K and anextension ν s of ν to K sep (or K ). Write G K = Gal ( K sep /K ) = Aut K ( K ) and G h = { σ ∈ G K | ν s ◦ σ = ν s } the decomposition group of ν s . The decomposition field of ν s which is by definition K h ( ν s ) = { x ∈ K sep | σ ( x ) = x, ∀ σ ∈ G h } is a henselization of ( K, ν ) . Any other choice for the extension ν s will give another henselization. All henselizationsare isomorphic up to unique isomorphism so we will talk about the henselization of ( K, ν ) and write it ( K h , ν h ) .In our situation we are interested in certain types of extensions. Once we have our truncated valuation µ Q , wewish to enumerate the valuations that extend both µ Q and ν , to K [ X ] . We use the following result concerningthe permutation of valuations by automorphisms of extensions. Consider ( K, ν ) a valued field and L/K a fieldextension. We will write E ( L, ν ) the set of valuations on L extending ν . Proposition 6.18.
Let
L/K be a normal field extension and ν a valuation of K . Aut K ( L ) acts transitively on theset of extensions of ν to L as follows Aut K ( L ) × E ( L, ν ) −→ E ( L, ν )( σ, µ ) µ ◦ σ − = µ σ . Once we have a common extension µ of our µ Q and ν , so for instance ν a,δ where ( a, δ ) is a minimal pair associatedto Q , then we can let G K = Aut K ( K ) act on µ so that we obtain all the extensions of µ Q . Indeed K ( X ) /K ( X ) isa normal extension with same Galois group as K/K . So, if σ ∈ G K , we have the following equivalences µ ◦ σ − ∈ E ( K [ X ] , ν ) ⇐⇒ µ ◦ σ − (cid:12)(cid:12) K = ν ⇐⇒ µ (cid:12)(cid:12) K ◦ σ − (cid:12)(cid:12) K = ν since σ − ( K ) ⊂ K ⇐⇒ ν ◦ σ − (cid:12)(cid:12) K = ν ⇐⇒ σ − (cid:12)(cid:12) K ∈ G h . Thus, if we identify the automorphism groups of K ( X ) /K ( X ) and K/K we can state the proposition below.
Proposition 6.19.
Take an abstract key polynomial Q and ( a, δ ) an associated minimal pair. We thus have E ( K [ X ] , ν ) ∩ E ( K [ X ] , µ Q ) = { ν a,δ ◦ σ − | σ ∈ G h } = { ν σ ( a ) ,δ | σ ∈ G h } . Proof.
It remains to prove the following identity ν a,δ ◦ σ − = ν σ ( a ) ,δ for any σ ∈ G h . For f ∈ K [ X ] decompose it as f = P i > c i ( X − σ ( a )) i ν a,δ ◦ σ − ( f ) = ν a,δ X i > σ − ( c i )( X − a ) i = min i > { ν ( σ − ( c i )) + iδ } = min i > { ν ( c i ) + iδ } = ν σ ( a ) ,δ ( f ) . Remark 6.20.
Just as G h acts on valuations, it also acts on balls. Set σ ∈ G h and ( a, δ ) ∈ K × Φ( ν ) . Then σ ( D ( a, δ )) = D ( σ ( a ) , δ ) . Indeed, for any x ∈ K , one has ν ( x − a ) = ν ( σ ( x ) − σ ( a )) .24 .4. Return to diskoids. We can use the action of G h to make the structure of diskoids easier to grasp and makemore sense of the maps ν e D ( f,µ ( f )) . Proposition 6.21.
Consider a polynomial f ∈ K h [ X ] , such that G h acts transitively on its roots and µ ( f ) ∈ Φ( ν ) .If we set a ∈ K an optimising root of f ( i.e., µ ( X − a ) = δ ( f ) ), then we have e D ( f, µ ( f )) = [ σ ∈ G h ( ν ) D ( σ ( a ) , δ ( f )) . Thus ν e D ( f,µ ( f )) is in fact a valuation on K [ X ] (or even K h [ X ] ) and its extensions to K [ X ] are ν D ( σ ( a ) ,δ ( f )) , σ ∈ G h . Proof.
Considering Lemma 6.9, the union is well-indexed, seeing as the roots of f will be σ ( a ) , σ ∈ G h . Wealready know ν D ( σ ( a ) ,δ ( f )) is well-defined valuation and equal to the valuation given by the defining pair ( a, δ ( f )) .Consider any g ∈ K h [ X ] , we will show that in fact ν D ( σ ( a ) ,δ ( f )) ( g ) = ν D ( a,δ ( f )) ( g ) this value thus being also equalto ν D ( f,µ ( f )) ( g ) . Indeed, for any σ ∈ G h , σ will not change the coefficients of g , thus we can safely write ν D ( σ ( a ) ,δ ( f )) ( g ) = ν σ ( a ) ,δ ( f ) ( g )= ν a,δ ( f ) ◦ σ − ( g )= ν a,δ ( f ) ( g )= ν D ( a,δ ( f )) ( g ) . Since ν D ( a,δ ( f )) is a valuation over K [ X ] , extending ν e D ( f,µ ( f )) , by Proposition 6.18, we know that all the otherextensions are ν a,δ ( f ) ◦ σ − = ν D ( σ ( a ) ,δ ( f )) .For instance if the polynomial f in the preceding theorem is irreducible, then it has its roots permuted transi-tively. Corollary 6.22.
Let Q be an abstract key polynomial with µ ( Q ) ∈ Φ( ν ) . If Q is irreducible over K h , then ν D ( Q,µ ( Q )) is a valuation and in fact µ Q = ν e D ( Q,µ ( Q )) . Proof.
By Theorem 5.3, if we assign to Q one of its minimal pairs ( a, δ ) µ Q = ν a,δ (cid:12)(cid:12) K [ X ] = ν D ( a,δ ) (cid:12)(cid:12) K [ X ] = ν e D ( Q,µ ( Q )) . We thus formulate the following concept.
Definition 6.23.
We say that a polynomial f ∈ K [ X ] is analytically irreducible if it is irreducible over K h If Q is analytically irreducible, we can then extend the correspondence of Theorem 6.4, into the one below. (cid:26) ABKPs Q with µ ( Q ) ∈ Φ( ν ) (cid:27) { RT Extensions } {
Diskoids } Qµ Q = ν D D = e D ( Q, µ ( Q )) ontoonto one-to-one
25f the base field ( K, ν ) is itself henselian, the answer is obviously affirmative (since ABKPs are irreducible over K to begin with), thus the correspondence is true in this case. If ( K, ν ) is of rank , we show that an ABKP Q with Φ( µ ) ⊆ Φ( ν ) ⊗ Q , i.e., µ ( Q ) ∈ Φ( ν ) ,is analytically irreducible.The reasoning can be done by reductio ad absurdum. Suppose on the contrary that Q is reducible in K h [ X ] . Wecan write Q = P R, P, R ∈ K h [ X ] with P irreducible over K h and ǫ ( P ) = ǫ ( Q ) . Indeed consider a an optimisingroot of Q and suppose P is its minimal polynomial over K h . Then P | Q and since Q is not irreducible, we get that P = Q .In rank one, any element in the henselization can be approached arbitrarily. Indeed, one can embed the valuedgroups Φ( ν ) ⊆ R and one knows that the completion of a valued field is henselian, thus it contains the henselization,so ∀ z ∈ K h , ∀ ǫ > , ∃ z ′ ∈ K, ν h ( z − z ′ ) > ǫ. We now use the continuity of roots of a monic polynomial of constant degree relative to its coefficients. We useas a reference [8, Theorem 2], but more specifically the remark that follows Theorem 2 of the aforementioned paper.We write V ( P l a l X l ) = min l ν ( a l ) so that we can state the theorem below. Theorem 6.24.
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