On irreducible components of real exponential hypersurfaces
OON IRREDUCIBLE COMPONENTS OF REAL EXPONENTIALHYPERSURFACES
CORDIAN RIENER AND NICOLAI VOROBJOVA
BSTRACT . Fix any real algebraic extension K of the field Q of rationals. Poly-nomials with coefficients from K in n variables and in n exponential functionsare called exponential polynomials over K . We study zero sets in R n of exponen-tial polynomials over K , which we call exponential-algebraic sets . Complementsof all exponential-algebraic sets in R n form a Zariski-type topology on R n . Let P ∈ K [ X , . . . , X n , U , . . . , U n ] be a polynomial and denote V : = { ( x , . . . , x n ) ∈ R n | P ( x , . . . , x n , , e x , . . . , e x n ) = } .The main result of this paper states that, if the real zero set of a polynomial P isirreducible over K and the exponential-algebraic set V has codimension 1, then,under Schanuel’s conjecture over the reals, either V is irreducible (with respect tothe Zariski topology) or each of its irreducible components of codimension 1 is arational hyperplane through the origin. The family of all possible hyperplanes isdetermined by monomials of P . In the case of a single exponential (i.e., when P isindependent of U , . . . , U n ) stronger statements are shown which are independentof Schanuel’s conjecture.
1. I
NTRODUCTION
The main motivation of this paper is to begin a study of irreducible components ofzero sets of functions defined by expressions that are polynomial in variables andexponentials of variables. The components are meant to be defined by the sametype of expressions. The interesting questions include: do the components haveany special structure and what is an upper bound on their number.Throughout this article we will denote tuples of variables by X : = ( X , . . . , X n ) and U : = ( U , . . . , U n ) , and with a given tuple X we associate the tuple of ex-ponential functions e X : = ( e X , . . . , e X n ) . We consider the field of real algebraicnumbers R alg , the field Q of rational numbers, and we fix a real algebraic exten-sion K of Q . Further, K [ X , U ] : = K [ X , . . . , X n , U , . . . , U n ] will denote the ring ofpolynomials with coefficients in K in the 2 n variables. Clearly, K [ X , e X ] is a ring offunctions which we call the ring of exponential polynomials over K (or E-polynomials ,for brevity). The geometry and model theory of zero sets of E -polynomials are wellunderstood (see for example [4, 7, 10]). In the special case when P is independentof variables X , the E -polynomial is called exponential sum (with integer spectrum).The theory of zero sets of exponential sums is also developed, in particular in[6, 18], but apparently not for the structure of their irreducible components.Every P ∈ K [ X , U ] defines an E -polynomial f via the map E : K [ X , U ] → K [ X , e X ] , Mathematics Subject Classification.
Key words and phrases.
Exponential-algebraic set, irreducible components, Schanuel’s conjecture. a r X i v : . [ m a t h . AG ] J u l CORDIAN RIENER AND NICOLAI VOROBJOV such that f ( X ) = E ( P ( X , U )) = P ( X , e X ) .A finite set of polynomials P : = { P , . . . , P k } ⊂ K [ X , U ] defines a real algebraicsetZer ( P ) : = { ( x , u ) = ( x , . . . , x n , u , . . . , u n ) ∈ R n | P ( x , u ) = · · · = P k ( x , u ) = } ,and similarly, we will denote by Zer ( f , . . . , f l ) ⊂ R n the zero set of a finite set of E -polynomials. We will call any Zer ( f , . . . , f l ) ⊂ R n a (real) exponential-algebraicset or just exponential set , for brevity. By taking the sum of squares, every real alge-braic set (respectively, every exponential set) can be defined as a zero set of a singlepolynomial (respectively, E -polynomial). An exponential set V will be called re-ducible (over K ) if there are two distinct non-empty exponential sets V , V suchthat V = V ∪ V , and irreducible otherwise. It will be shown in Section 2 thatevery exponential set can be uniquely represented as a finite union of irreducibleexponential subsets (called irreducible components) neither of which is containedin another.In this article we will be concerned with the structure of irreducible componentsand will prove the following main result. Theorem 1.1.
Let P ∈ K [ X , U ] and assume that Zer ( P ) ⊂ R n is an irreducible realalgebraic set. Further, let f = E ( P ) and assume that the dimension (see Definition 2.12below) of Zer ( f ) is n − . Then, assuming Schanuel’s conjecture, either Zer ( f ) is alsoirreducible, or every of its ( n − ) -dimensional irreducible components is a rational hy-perplane through the origin. Schanuel’s conjecture is formulated at the beginning of Section 5. In the case of asingle exponential term, i.e., when P is independent of all, but possibly one, vari-ables U , . . . , U n , the theorem can be made stronger and independent of Schanuel’sconjecture (see Theorem 3.5 below).Let us illustrate Theorem 1.1 by some examples. Example 1.2.
Let P = X − U +
1. The straight line Zer ( P ) ⊂ R is, of course,irreducible. According to Theorem 1.1, the exponential set V = Zer ( X − e X + ) (consisting of two points) is irreducible. This can also be seen by the following ele-mentary argument, independent of Schanuel’s conjecture, which in various mod-ifications is used throughout this paper. Suppose V is reducible. The only way tosplit two-point set into two distinct non-empty parts is the partition into single-tons. Then the non-zero point A ∈ V can be defined as A = Zer ( Q ( X , e X )) forsome Q ( X , U ) ∈ K [ X , U ] . It follows that A is the projection along U of an isolatedpoint in Zer ( P , Q ) with algebraic coordinates. This contradicts the Lindemanntheorem. Example 1.3.
Let P : = X U + X U − X − X and f : = E ( P ) . Then dim Zer ( P ) = ( f ) =
1. The polynomial P is irreducible over K (even over R ),hence the algebraic set Zer ( P ) is irreducible over K . On the other hand, Zer ( f ) ⊂ R is reducible and consists of two irreducible components, which are the linesZer ( X ) and Zer ( X ) . Example 1.4.
This example illustrates the case when for an irreducible Zer ( P ) theexponential hypersurface Zer ( E ( P )) has irreducible components of codimensiongreater than 1. RREDUCIBLE COMPONENTS 3 (a) The transformed Cartan umbrella (b) The intersection with e X = U in the line L andthe point A F IGURE
1. Visualization of Example 1.4Consider the polynomial P : = ( X + U − )(( X − U + ) + X ) + ( X − U + ) .Note that Zer ( P ) is an affine transformation the Cartan umbrella [3]. The algebraicset Zer ( P ) ⊂ R contains the straight line L : = Zer ( U − X ) , therefore, it inter-sects with the surface Zer ( U − e X ) along this line. It also contains the straight lineZer ( X − U + X } , which intersects with Zer ( U − e X ) by exactly two points, (
0, 0, 1 ) ∈ L and another point, A , with transcendental coordinates. We now provethat Zer ( P ) ∩ Zer ( U − e X ) = L ∪ { A } .Let f ( X , X ) : = P ( X , X , e X ) . Note that X + e X − X =
0, hence for X (cid:54) = f = X = − ( X − e X + ) (cid:18) X − e X + X + e X − + (cid:19) .Now we prove that(1.2) 2 X − e X + X + e X − > − X (cid:54) =
0. If X > X + e X − >
0, hence (1.2) is equivalent to2 X − e X + > − X − e X + X >
0, which is obviously true. If X <
0, then X + e X − <
0, so (1.2)is equivalent to 2 X − e X + < − X − e X + X <
0, which again is true. It follows that (1.1) can hold true (for X (cid:54) =
0) ifand only if X = X − e X + =
0, and our claim is proved.
CORDIAN RIENER AND NICOLAI VOROBJOV
The polynomial P is irreducible over R , hence the algebraic set Zer ( P ) ⊂ R is ir-reducible over K . On the other hand, the set Zer ( f ) ⊂ R is reducible over K , withtwo irreducible components: one-dimensional Zer ( X ) and zero-dimensionalZer ( X − e X + X ) which consists of two points, rational (
0, 0 ) and transcendental projection of A along U . 2. R EGULAR AND SINGULAR POINTS
The aim of this section is to prove that taking Zariski closure of a set of all points ofa given local dimension does not increase the dimension (Theorem 2.19). The ideais to consider the singular locus of the ambient exponential set. To that end, weadjust to the exponential case the standard routine of definitions and statementsregarding regular and singular points. We note some differences with the algebraiccase, in particular that the dimension of a singular locus of an exponential set mayremain the same as the dimension of the set.Recall that in the introduction we defined the map E : K [ X , U ] → K [ X , e X ] ,such that E ( P ( X , U )) = P ( X , e X ) , for every P ∈ K [ X , U ] . Lemma 2.1.
The map E is an isomorphism of rings.Proof.
It is immediate that E is an epimorphism of rings. Thus, it remains to es-tablish that E is injective. In order to argue by contradiction, assume that E is notinjective. Thus, there exist distinct P , Q ∈ K [ X , U ] such that the functions E ( P ) and E ( Q ) coincide. Since P and Q are distinct, the algebraic set Zer ( P − Q ) hasdimension at most 2 n −
1. Also, Zer ( U − e X ) ⊂ Zer ( P − Q ) . Observe that the di-mension of the Zariski closure Z of { x ∈ R n | { x } × R n ⊂ Zer ( P − Q ) } is less than n . Take a point x = ( x , . . . , x n ) ∈ R nalg \ Z such that its coordinates are linearlyindependent over Q . We have that ( x , e x ) ∈ Zer ( U − e X ) ⊂ Zer ( P − Q ) , hence thenumbers x , . . . , x n , e x , . . . , e x n are algebraically dependent over Q . But now it fol-lows from the Lindemann-Weierstrass theorem that numbers x , . . . x n are linearlydependent over Q which contradicts the choice of x . (cid:3) Lemma 2.1 immediately implies the following corollary.
Corollary 2.2.
The ring K [ X , e X ] is Noetherian. Given an exponential set V ⊂ R n , let I ( V ) ⊂ K [ X , e X ] denote the set of all E -polynomials in K [ X , e X ] that vanish on V . It is easy to see that I ( V ) is an ideal in K [ X , e X ] .The following corollary is an immediate implication of Noetherianity. Corollary 2.3.
The ideal I ( V ) is finitely generated. The next corollary is a standard implication of Noetherianity (see [5, Proposi-tion 1.1]).
Corollary 2.4.
The complements of all exponential sets in R n form a Noethrian topologyon R n . We call this topology the Zariski topology . RREDUCIBLE COMPONENTS 5
Definition 2.5.
A non-empty subset Y of a topological space X is called irreducible if it cannot be represented as a union Y = Y ∪ Y of its proper subsets Y and Y each of which is closed in Y . For an empty set Y irreducibility is undefined.Applying this definition to X = R n , equipped with the Zariski topology definedin Corollary 2.4, and Y being an exponential set, we get the definition of an irre-ducible exponential set.The following corollary is another standard implication of Noetherianity (see [5,Proposition 1.5]). Corollary 2.6.
Every non-empty exponential set V ⊂ R n can be represented as a finiteunion V = V ∪ · · · ∪ V t of irreducible exponential sets V i . If we require V i (cid:54)⊂ V j fori (cid:54) = j, then the sets V i are defined uniquely and are called irreducible components of V. We borrow the definition of a regular point of an exponential set from real alge-braic geometry (see [2, Definition 3.2.2]). Let V ⊂ R n be an exponential set and I ( V ) = ( f , . . . , f k ) its ideal generated by exponential polynomials f , . . . , f k ∈ K [ X , e X ] . Set r = sup x ∈ V rank ( V , x ) , whererank ( V , x ) : = rank (cid:18) ∂ ( f , . . . , f k ) ∂ ( X , . . . , X n ) ( x ) (cid:19) .Note (see [2]) that the number r does not depend on the choice of the set of gen-erators f , . . . , f k . The number r is called the rank of the ideal I ( V ) , and we write r = rank I ( V ) . Definition 2.7.
Let V ⊂ R n be an exponential set.(1) If V is irreducible then a point x ∈ V is called a regular point of V ifrank ( V , x ) = rank I ( V ) .(2) If V is reducible then x ∈ V is called regular point of V if there exists oneand only one irreducible component of V containing x .(3) A point x ∈ V that is not regular it is called a singular point of V .(4) We denote the subset of all regular points of V by Reg ( V ) and the subsetof all singular points of V by Sing ( V ) .(5) An exponential set is called non-singular if V = Reg ( V ) .The proof of the following lemma is exactly the same as the proof of Proposi-tion 3.2.4 in [2] for algebraic sets. Lemma 2.8.
The set
Sing ( V ) is an exponential set properly contained in V. Lemma 2.9.
Let V ⊂ R n be an exponential set. Then there is a finite filtration (2.1) V ⊃ Sing ( V ) ⊃ Sing ( Sing ( V )) ⊃ Sing ( Sing ( Sing ( V ))) ⊃ · · · such that the last set in the chain is non-singular.Proof. Termination of the chain follows from Noetherianity since for every expo-nential set V ⊂ R n the set Sing ( V ) is again an exponential set. The last set in thischain is non-singular because otherwise the set of its singular points would definea proper subset. (cid:3) Remark . It follows from Lemma 2.9 that we can define a “non-singular strati-fication” of an exponential set V ⊂ R n as the following finite partition V = Reg ( V ) ∪ Reg ( Sing ( V )) ∪ Reg (( Sing ( Sing ( V )))) ∪ · · · . CORDIAN RIENER AND NICOLAI VOROBJOV
The proof of the following lemma is exactly the same as the proof of Proposi-tion 3.2.9 in [2] for algebraic sets.
Lemma 2.11.
If V ⊂ R n is an irreducible exponential set, then for every x ∈ Reg ( V ) there exists a Zariski neighbourhood U of x in R n , and exponential polynomials f , . . . , f r ∈ I ( V ) such that V ∩ U = U ∩ { f = · · · = f r = } , and rank (cid:18) ∂ ( f , . . . , f r ) ∂ ( X , . . . , X n ) ( y ) (cid:19) = rank I ( V ) = rfor every y ∈ V ∩ U.In particular,
Reg ( V ) is a real analytic submanifold of R n of dimension n − r. Remark 2.10 and Lemma 2.11 allow to define the notion of the dimension of anexponential set.
Definition 2.12.
Let V ⊂ R n be an exponential set.(1) The dimension of V , denoted by dim V , is the maximal dimension of realanalytic manifolds comprising the non-singular stratification of V . Thedimension dim ( Reg ( V )) of the set of regular points of V is its dimensionas a real analytic manifold.(2) If dim V = n −
1, we call V a hypersurface in R n .(3) For x ∈ V we denote by dim x V the local dimension of V at x , i.e., the max-imal dimension of real analytic manifolds which are intersections of ele-ments of the non-singular stratification of V with an Euclidean neighbour-hood of x in R n .For an exponential set V ⊂ R n one can also introduce the analogy of the Krulldimension as follows. Definition 2.13.
The
Krull dimension of V , denoted by dim K ( V ) , is largest d ∈ N such that there is a filtration V ⊂ V ⊂ · · · ⊂ V d of pair-wise distinct irreducibleexponential subsets of V .Unlike the case of real or complex algebraic sets, dim ( V ) does not necessarily coin-cide with dim K ( V ) as is shown in the following example. Example 2.14.
Consider the irreducible two-point exponential set V : = Zer ( X − e X + ) ⊂ R from Example 1.2. Observe that T : = Zer ( X ) is also irreducible. Since we have T (cid:40) V it follows that dim K V =
1, while dim V =
0. Further, we can concludefrom dim K V = K R ≥
2. On the other hand dim K R ≤
2, since everynonempty irreducible exponential subset in R , consisting of a finite number ofpoints, having a point different from T , and properly containing another such set,would contain an algebraic point different from T , which contradicts Lindemann’stheorem. So we conclude that dim K R =
2. Note that every exponential set in R which is irreducible over R (rather than over K ) is an irreducible algebraic set over R (actually, a singleton). Hence, the Krull dimension of R is 1 in this case.In view of this example we emphasize that in the sequel we will be using theconcept of dimension exclusively in the sense of Definition 2.12.We can also observe that unlike the case of real or complex algebraic sets, the di-mension of an exponential set V may coincide with the dimension of its singularlocus Sing ( V ) as shown in the following example. RREDUCIBLE COMPONENTS 7
Example 2.15.
Let V : = Zer (cid:0) ( X − e X + )( X − e X + ) (cid:1) ⊂ R . Clearly, T : = Zer ( X ) ⊂ Sing ( V ) (in fact, T = Sing ( V ) ) but dim ( T ) = dim ( V ) . Lemma 2.16.
For any exponential set V one has dim ( Sing ( V )) ≤ dim ( Reg ( V )) .Proof. Assume first that V is irreducible. We prove the statement by induction onthe length of the filtration (2.1). If (2.1) consists of a single exponential set, V , thenthis set is non-singular, so the base of induction is true.Since, by Lemma 2.8, I ( Sing ( V )) ⊃ I ( V ) , we conclude thatrank I ( Sing ( V )) ≥ rank I ( V ) .By Lemma 2.11, dim ( Reg ( V )) = n − rank ( I ( V )) and dim ( Reg ( Sing ( V ))) = n − rank ( I ( Sing ( V )) ) ,hence(2.2) dim ( Reg ( V )) ≥ dim ( Reg ( Sing ( V ))) .By the inductive hypothesis,dim ( Reg ( Sing ( V ))) ≥ dim ( Sing ( Sing ( V ))) ,which, together with the previous inequality, implies that(2.3) dim ( Reg ( V )) ≥ dim ( Sing ( Sing ( V ))) .Since dim ( Sing ( V )) = max { dim ( Reg ( Sing ( V ))) , dim ( Sing ( Sing ( V ))) } ,we conclude from (2.2) and (2.3) that dim ( Reg ( V )) ≥ dim ( Sing ( V )) .Now suppose that V is reducible and V ( ) , V ( ) are two of its irreducible compo-nents. Let V ( ) = V ( ) ∩ V ( ) . Since dim (cid:16) V ( ) (cid:17) ≤ min { dim (cid:16) V ( ) (cid:17) , dim (cid:16) V ( ) (cid:17) } and, by the first half of the proof,dim (cid:16) V ( ) (cid:17) = dim (cid:16) Reg (cid:16) V ( ) (cid:17)(cid:17) , dim (cid:16) V ( ) (cid:17) = dim (cid:16) Reg (cid:16) V ( ) (cid:17)(cid:17) ,we get dim (cid:16) V ( ) (cid:17) ≤ dim ( Reg ( V )) . (cid:3) Corollary 2.17.
Let dim ( V ) = r and dim x ( V ) < r for some x ∈ V. Then x ∈ Sing ( V ) .Proof. Let, contrary to the claim, x ∈ Reg ( V ) . By Lemma 2.11, dim x ( V ) is thesame at every point in Reg ( V ) . Since, by Lemma 2.16, dim Reg ( V ) = dim ( V ) ,we conclude that dim x ( V ) = r which is a contradiction. (cid:3) Definition 2.18.
For an exponential set V and 0 ≤ p ≤ dim ( V ) denote V p : = { x ∈ V | dim x ( V ) = p } . Theorem 2.19.
Let V p (cid:54) = ∅ for some ≤ p ≤ dim ( V ) . There is an exponential subsetW ⊂ V such that dim ( W ) = p and V (cid:96) ⊂ W for all ≤ (cid:96) ≤ p. CORDIAN RIENER AND NICOLAI VOROBJOV
Proof.
We prove that W can be found among the sets of the filtration defined in(2.1). If p = dim ( V ) we take W = V . Let p < dim ( V ) . By Corollary 2.17, V p ⊂ Sing ( V ) . Passing from an exponential set S i in the chain (2.1) to the nextone on the right, S i + , consists of removing from S i an equidimensional subsetReg ( S i ) having the highest dimension, dim ( S i ) . On the other hand, (2.1) cannotconsist only of sets of dimension strictly greater than p , since the last set in (2.1) isa non-singular equidimensional set, while V p (cid:54) = ∅ . Hence, there is the first (fromthe left) set in the filtration having the dimension p , which can be taken as W . (cid:3) Remark . For an integer k such that 0 ≤ k ≤ n , let X k : = ( X , . . . , X k ) , U k : = ( U , . . . , U k ) , and e X k : = ( e X , . . . , e X k ) .All definitions in this section can be extended to the ring of functions K [ X , e X k ] (polynomials in n variables X , . . . , X n and k exponentials e X , . . . , e X k ). In partic-ular, we consider Zariski topology in R n with all closed sets of the kind Zer ( f ) ,where f ∈ K [ X , e X k ] . If Zer ( f ) is irreducible with respect to this topology, we willsay that it is irreducible in K [ X , e X k ] . It is easy to check that all statements in thissection, including Theorem 2.19, hold true for an arbitrary fixed k .3. C ASE OF A SINGLE EXPONENTIAL
In this section we consider the case of exponential sets that involve only a singleexponential, i.e., exponential sets defined by E -polynomials f = P ( X , . . . , X n , e X ) with P ∈ K [ X , U ] . We denote V : = Zer ( f ) and m : = dim ( V ) . Let π : R n + → R n be the projection map along U . Definition 3.1.
A real algebraic set W ⊂ R n + is called admissible for V if V = π (cid:16)(cid:16) W ∩ Zer (cid:16) U − e X (cid:17)(cid:17) ∪ Zer ( P , X , U − ) (cid:17) and dim ( W ) ≤ m + Lemma 3.2.
There exists an admissible set for V.Proof.
A proof of the statement immediately follows from [13, Section 7].Alternatively, if the set(3.1) { (cid:96) ∈ Z | ≤ (cid:96) ≤ m + ( Zer ( P ) \ Zer ( X )) (cid:96) (cid:54) = ∅ } is non-empty, then let r denote its maximal element. By Theorem 2.19 (or its easierversion for algebraic sets), there is an r -dimensional algebraic set W ⊂ Zer ( P ) containing the semialgebraic set ( Zer ( P )) r as well as all sets ( Zer ( P )) (cid:96) where 0 ≤ (cid:96) ≤ r . If the set (3.1) is empty, assume W = ∅ . Then W is an admissible set for V ,according to Theorem 7.2 in the Appendix. (cid:3) Lemma 3.3.
Let W ( ) , . . . , W ( t ) be all irreducible components of an admissible set W forV. Let A be an m-dimensional irreducible component of V. Then, either A coincides with π ( W ( i ) ∩ Zer ( U − e X )) for some ≤ i ≤ t, or A is the union of an m-dimensionalalgebraic subset of Zer ( X ) and a (possibly empty) set of points having local dimensionsless than m.Proof. Since A is irreducible, it is either a subset of Zer ( X ) or a subset of the pro-jection π (( W ( i ) ∩ Zer ( U − e X )) for a certain 1 ≤ i ≤ t . In the first case the proofis completed. So assume that A ⊂ π ( W ( i ) ∩ Zer ( U − e X )) . If A = π ( W ( i ) ∩ RREDUCIBLE COMPONENTS 9
Zer ( U − e X )) , then the proof is completed. Suppose now that A (cid:54) = π ( W ( i ) ∩ Zer ( U − e X )) . Then there exists another irreducible component, B , of V suchthat B ⊂ π ( W ( i ) ∩ Zer ( U − e X )) . Let W ( i ) = Zer ( R ) and A = π ( Zer ( Q , U − e X )) for some polynomials R , Q ∈ K [ X , U ] . Thus, there are two algebraic sets, W ( i ) = Zer ( R ) and Zer ( Q ) with a non-empty intersection, and Zer ( R ) is irre-ducible. Since we have B (cid:54)⊂ π ( Zer ( Q )) it follows that Zer ( R ) (cid:54)⊂ Zer ( Q ) . Then,dim ( Zer ( R , Q )) < dim ( Zer ( R )) , hence dim ( Zer ( R , Q )) = m . Therefore, the set S : = Zer ( Q , U − e X ) is an m -dimensional real analytic subset of m -dimensionalalgebraic set Zer ( R , Q ) .By Lindemann’s theorem, the set S \ Zer ( U − X ) does not contain points withalgebraic coordinates. Hence, dim x S < m at every x ∈ S \ Zer ( U − X ) . Itfollows that A is the union of an algebraic set π ( Zer ( Q , U − X )) and a set ofpoints having local dimensions less than m . (cid:3) Corollary 3.4.
Let
Zer ( P ) be an irreducible algebraic set and let dim ( V ) = m = n − .Then V is either irreducible or every ( n − ) -dimensional irreducible component of V isthe union of the hyperplane Zer ( X ) and a set of points having local dimensions less thann − .Proof. We can choose the set Zer ( P ) as an admissible set for V , and then applyLemma 3.3. (cid:3) Theorem 3.5.
Every m-dimensional irreducible component of V either coincides with π ( W ( i ) ∩ Zer ( U − e X )) for some ≤ i ≤ t, or is an algebraic subset of Zer ( X ) .Proof. It follows from Lemma 3.3 that if an irreducible component A of V does notcoincide with one of π ( W ( i ) ∩ Zer ( U − e X )) , it is the union of a m -dimensionalalgebraic subset of Zer ( X ) and a set B of points having local dimensions not ex-ceeding p < m . Let C : = B \ Zer ( X ) , then C ⊂ V p . To argue by contradictionsuppose that C (cid:54) = ∅ . According to Theorem 2.19, there is an exponential sub-set T ⊂ V such that dim ( T ) = p and V p ⊂ T . Hence, A can be represented asthe union of two distinct non-empty exponential sets, A ∩ Zer ( X ) and A ∩ T . Itfollows that A is reducible, which is a contradiction. (cid:3) Example 3.6.
Let P : = X + ( X + ( U − ) − ) . Note that Zer ( P ) ⊂ R is a 1-dimensional set (a unit circle, centered at (
0, 0, 1 ) ) in the coordinate plane Zer ( X ) ,hence is irreducible. Let V = Zer ( E ( P )) . We can choose the admissible fam-ily for V consisting of the unique set Zer ( P ) . Then V is the projection of the0-dimensional algebraic set Zer ( P , U − X ) , is reducible, and consists of twoirreducible components, Zer ( X + ( X + ) ) and Zer ( X + ( X − ) ) .The following corollary is immediate. Corollary 3.7. If Zer ( P ) is an irreducible algebraic set and dim ( V ) = n − , then V iseither irreducible or it has a unique ( n − ) -dimensional irreducible component coincidingwith the hyperplane Zer ( X ) . For the case of a reducible V this corollary can be illustrated by Example 1.4. Corollary 3.8.
The number of all irreducible components of V does not exceed ( cd ) n ,where c is an absolute positive constant and d is the total degree of polynomial P. Proof.
Theorem 1 in [14] implies that the sum of numbers of all (absolutely) ir-reducible components of Zariski closures of sets ( Zer ( P )) (cid:96) over all (cid:96) =
0, . . . , m does not exceed ( c d ) n for an absolute positive constant c . Also the number ofall irreducible components of the algebraic set Zer ( P , U − X ) does not exceed ( c d ) n for an absolute positive constant c . Now the corollary follows from Theo-rem 3.5. (cid:3) Observe that the bound in Corollary 3.8 is asymptotically tight because it is tightalready for polynomials.4. C
ASE OF MANY EXPONENTIALS
Consider a polynomial P ∈ K [ X , U ] . Then every monomial of P , with respect tothe variables U , . . . , U n , is of the kind A ν U d ν · · · U d n ν n with A ν ∈ K [ X ] , d i ν ≥ P the following union of linear subspaces: W P : = (cid:91) ν , µ { d ν X + · · · + d n ν X v = d µ X + · · · + d n µ X n } ,where the union is taken over all pairs of different monomials. (If there is at mostone monomial with respect to U , then W P is undefined.)The following lemma is a version of the Lindemann-Weierstrass theorem. Lemma 4.1.
If for some point x = ( x , . . . , x n ) ∈ R nalg the polynomial Q : = P ( x , U ) ∈ R alg [ U ] is not identically zero and Q ( e x , . . . , e x n ) = , then x ∈ W P .Proof. This is a slight adjustment of a standard proof of the Lindemann-Weierstrasstheorem.We have: Q ( e x , . . . , e x n ) = ∑ ν A ν ( x ) e d ν x + ··· + d n ν x n = A ν ( x ) are not all zero. Removing all terms with zero coef-ficients, assume that in this sum all coefficients are non-zero. Obviously, at leasttwo terms will remain, one of which may be a non-zero constant. By Baker’s re-formulation of the Lindemann-Weierstrass theorem [1, Theorem 1.4], the powers d ν x + · · · + d n ν x n are not pair-wise distinct. It follows that x ∈ W P . (cid:3) Denote f : = E ( P ) , V : = Zer ( f ) ⊂ R n , V (cid:48) : = V \ W P . Lemma 4.2.
V is an algebraic set if and only if V (cid:48) × R n ⊂ Zer ( P ) .Proof. The “if” part of the implication is trivial.Suppose now that V is algebraic and V (cid:48) (cid:54) = ∅ . Observe that the set { x ∈ V (cid:48) | { x } × R n ⊂ Zer ( P ) } is closed in V (cid:48) (with respect to the Euclidean topology). Hence, thecomplement V (cid:48)(cid:48) of this set in V (cid:48) is open in V (cid:48) . Suppose that contrary to the claim, V (cid:48) × R n (cid:54)⊂ Zer ( P ) , i.e., V (cid:48)(cid:48) (cid:54) = ∅ . The algebraic points in V are everywhere densein V since, by the assumption, V is an algebraic set. Therefore, there is a point x = ( x , . . . , x n ) ∈ V (cid:48)(cid:48) ∩ R nalg such that the polynomial Q : = P ( x , U ) ∈ R alg [ U ] is not identically zero. Since ( e x , . . . , e x n ) ∈ Zer ( Q ) , we conclude, by Lemma 4.1,that x ∈ W P . This contradicts the choice of x . (cid:3) Corollary 4.3.
Let V be an irreducible algebraic set with dim ( V ) = n − . Then eitherV is a hyperplane in W P , or V × R n is an irreducible component of Zer ( P ) . RREDUCIBLE COMPONENTS 11
Proof. If V contains a hyperplane in W P , then it coincides with this hyperplane,since V is an algebraic set. If V is not a hyperplane in W P , then dim ( V ∩ W P ) < n −
1. Hence V (cid:48) n − (cid:54) = ∅ . Since V × R n is an irreducible algebraic set while, byLemma 4.2, dim ( V × R n ∩ Zer ( P )) = n −
1, we conclude that V × R n is an irre-ducible component of Zer ( P ) . (cid:3) Consider a polynomial S ∈ K [ X , U ] . Let g : = E ( S ) , T : = Zer ( g ) . Suppose thatdim ( T ) = n −
1, and that T n − ⊂ B , where B is an ( n − ) -dimensional algebraicset defined over K . Represent S as a polynomial in U with coefficients in K [ X ] .Then every monomial is of the kind A ν U d ν · · · U d n ν n , with A ν ∈ K [ X ] , d i ν ≥ W S : = (cid:91) ν , µ { d ν X + · · · + d n ν X n = d µ X + · · · + d n µ X n } ,where the union is taken over all pairs of different monomials. Lemma 4.4.
With the notations described above, the following inclusions take place:(i) ( T \ W S ) n − × R n ⊂ Zer ( S ) ;(ii) ( T \ ( W P ∪ W S )) n − × R n ⊂ Zer ( P , S ) ;(iii) ( T ∩ W S ) \ W P ) n − × R n ⊂ Zer ( P ) .Proof. We will only prove item (i), since the proofs of the other items are essentiallythe same.Denote A : = ( T \ W S ) n − . Suppose that A (cid:54) = ∅ . Observe that the set { x ∈ A | { x } × R n ⊂ Zer ( S ) } is closed in A . Thus, the complement C of this set in A isopen in A . Suppose that contrary to the claim, A × R n (cid:54)⊂ Zer ( S ) , i.e., C (cid:54) = ∅ . Thealgebraic points in T n − are everywhere dense in T n − since T n − ⊂ B . Therefore,there is a point x = ( x , . . . , x n ) ∈ C ∩ R nalg such that polynomials Q : = S ( x , U , . . . , U n ) ∈ R alg [ U ] are not identically zero. Since ( e x , . . . , e x n ) ∈ Zer ( Q ) , we conclude, by Lemma 4.1,that x ∈ W S . This contradicts the choice of x . (cid:3) Now we assume that V = Zer ( f ) ⊂ R n and T = Zer ( g ) ⊂ R n are exponentialsets, not necessarily algebraic. We will associate with these sets polynomials P , S and sets W P , W S as above. Lemma 4.5.
Let dim ( V ) = n − and Zer ( P ) be irreducible. Let T ⊂ V be a ( n − ) -dimensional irreducible component of V, with irreducible Zer ( S ) , such that there is an ( n − ) -dimensional algebraic set B (defined over K ) containing T n − . Then either V = Tor T n − ⊂ W P .Proof. Let A : = ( T \ ( W P ∪ W S )) n − . Suppose first that A (cid:54) = ∅ . By Lemma 4.4 (ii), A × R n ⊂ Zer ( S ) and A × R n ⊂ Zer ( P ) . Because dim ( A × R n ) = n −
1, and thesets Zer ( S ) , Zer ( P ) are irreducible algebraic, these sets coincide. It follows that T = V .Now suppose that A = ∅ and dim (( T ∩ W S ) \ W P ) = n − ( T ∩ W S ) n − consists of some hyperplanesin W S which are not all in W P , hence ( T ∩ W S ) \ W P contains a hyperplane, say L ,in R n . By Lemma 4.4 (iii), ( T ∩ W S ) \ W P ) n − × R n ⊂ Zer ( P ) . Therefore, Zer ( P ) contains a hyperplane L × R n , hence Zer ( P ) (being irreducible algebraic set) coin-cides with L × R n . It follows that both T and V coincide with the same hyperplane, L , in R n , thus again, T = V .If neither of the above alternatives take place, we have T n − ⊂ W P . (cid:3) Remark . Lemma 4.5 can be viewed as an analogy, in codimension 1, of the clas-sical Ax-Lindemann theorem (see its modern treatment in [11, Section 6]) whichdeals with exponential sums over complex numbers.
Corollary 4.7.
Let V ⊂ R n be an ( n − ) -dimensional algebraic set over K , which isirreducible as an algebraic set. Then it’s irreducible.Proof. Let T be an ( n − ) -dimensional irreducible exponential component of V .Then by Lemma 4.5, either T = V , or T n − ⊂ W P . In the former case we are done.In the latter case, by analytic continuation, T n − contains a hyperplane. There-fore, V also contains this hyperplane, moreover, being an irreducible algebraic set,coincides with this hyperplane. It follows that T = V . (cid:3)
5. P
ROOF OF THE MAIN THEOREM
Schanuel’s conjecture over real numbers is the following statement.
Suppose that for real numbers x , . . . , x n the transcendence degree td Q ( x , . . . , x n , e x , . . . , e x n ) < n . Then there are integers m , . . . , m n , not all zero, such that m x + · · · + m n x n = . This statement (along with its other versions) is the central, yet unsettled, conjec-ture in transcendental number theory (see [9, 8]).Throughout this section we will assume that for P ∈ K [ X , U ] the real algebraicset Zer ( P ) ⊂ R n is irreducible, and that for f = E ( P ) the exponential set V : = Zer ( f ) ⊂ R n , is a hypersurface, i.e., dim ( V ) = n −
1. The case n = n > Lemma 5.1.
Assuming Schanuel’s conjecture, every ( n − ) -dimensional irreduciblecomponent of V either coincides with V (V is irreducible), or it is the finite union ofhyperplanes through the origin, defined over Q , and a set of points having local dimensionless than n − .Proof. Suppose V is reducible and T is its irreducible component having dimen-sion n −
1. Then T = Zer ( g ) ⊂ R n for a suitable E -polynomial g such that g = E ( S ) , where S ∈ K [ X , U ] .Let dim Zer ( P ) = m for some n − ≤ m ≤ n −
1, and dim ( Zer ( P , S )) = (cid:96) forsome n − ≤ (cid:96) . Observe that (cid:96) < m , otherwise Zer ( P ) ⊂ Zer ( S ) since Zer ( P ) isirreducible, which contradicts the existence of components of V different from T .In particular, n ≤ m and (cid:96) ≤ n − ( n − ) -dimensional setZer ( P , S , U − e X , . . . , U n − e X n ) = Zer ( S , U − e X , . . . , U n − e X n ) to a coordinate subspace of some n − X , . . . , X α − , X α + , . . . , X n ,where 1 ≤ α ≤ n , is ( n − ) -dimensional. Consider any such α . Then the projec-tion contains a dense (in this projection) set of points ( x , . . . , x α − , x α + , . . . , x n ) ∈ RREDUCIBLE COMPONENTS 13 R n − alg and for each such point the intersectionZer ( S , P , X − x , . . . , X α − − x α − , X α + − x α + , . . . , X n − x n ) is an algebraic set defined over K .Observe that the set of points ( x , . . . , x α − , x α + , . . . , x n ) such that the dimensiondim ( Zer ( S , P , X − x , . . . , X α − − x α − , X α + − x α + , . . . , X n − x n )) is larger than (cid:96) − n + R n − having dimension less than n −
1. Hence, for a dense subset of algebraic points ( x , . . . , x α − , x α + , . . . , x n ) in R n − the dimension of the algebraic setZer ( S , P , X − x , . . . , X α − − x α − , X α + − x α + , . . . , X n − x n ) is at most (cid:96) − n +
1, i.e., at most n − P as a polynomial in U with coefficients in K [ X ] . Every monomial isthen of the kind A ν U d ν · · · U d n ν n ,with A ν ∈ K [ X ] , d j ν ≥
0. Consider the real algebraic set B : = (cid:91) ν { A ν = } ∪ (cid:91) ν , µ { d ν X + · · · + d n ν X n = d µ X + · · · + d n µ X n } ,where the first union is taken over all monomials, while the second union is takenover all pairs of different monomials.Suppose first that dim ( T \ B ) < n −
1. Then T n − ⊂ B . By Lemma 4.5, either V = T or T n − ⊂ W P . The first of these alternatives contradicts the reducibilityof V , hence, T n − is a union of rational hyperplanes through the origin, and thelemma is proved.Suppose now that dim ( T \ B ) = n −
1. Then there exists a number α , 1 ≤ α ≤ n ,a point ( x , . . . , x α − , x α + , . . . , x n ) ∈ R n − alg , and a number x α ∈ R such that(1) ( x , . . . , x n ) ∈ T \ B ;(2) the numbers x j , where j ∈ {
1, . . . , α − α +
1, . . . , n } , are linearly indepen-dent over Q ;(3) the dimension ofZer ( S , P , X − x , . . . , X α − − x α − , X α + − x α + , . . . , X n − x n ) is at most n − ( x , . . . , x α − , x α + , . . . , x n ) be such point. Then the setZer ( S , U − e X , . . . , U n − e X n ) ⊂ R n contains a point, namely, ( x , . . . , x n , e x , . . . , e x n ) , which also lies in an algebraicset Zer ( S , P , X − x , . . . , X α − − x α − , X α + − x α + , . . . , X n − x n ) of dimension at most n −
1. By Schanuel’s conjecture, m x + · · · + m n x n = m , . . . , m n , not all equal to 0. Since ( x , . . . , x α − , x α + , . . . , x n ) arealgebraic numbers, linearly independent over Q , we have m α (cid:54) =
0, hence, x α is alsoalgebraic. Thus, the point ( x , . . . , x n ) has real algebraic coordinates. Then ( x , . . . , x n ) ∈ B ,either because all coefficients A ν vanish (hence the polynomial P ( x , . . . , x n , U ) isidentically zero with respect to U ), or otherwise, by Lemma 4.1, since ( e x , . . . , e x n ) ∈ Zer ( S ( x , . . . , x n , U )) .This contradicts condition (1). It follows that components T , with the propertydim ( T \ B ) = n − (cid:3) Remark . In the proof of Lemma 5.1 the following implication of Schanuel’sconjecture was actually used, rather than Schanuel’s conjecture per se . If numbers x , . . . , x α − , x α + , . . . , x n ∈ R alg are linearly independent over Q , x α ∈ R , and the transcendence degree of x , . . . , x n , e x , . . . , e x n is less than n , then x α ∈ R alg . It is not known whether this particular case ofSchanuel’s conjecture is true. As M. Waldschmidt pointed out [17], this particularcase implies, for n =
2, that e and log 2 are algebraically independent, which is notknown. Proof of Theorem 1.1.
Suppose that V : = Zer ( f ) is reducible and T : = Zer ( g ) is itsirreducible component. Then, according to Lemma 5.1, T is the union of the set T ( ) of rational hyperplanes through zero, and a set T ( ) of points of some localdimensions less than n −
1. Suppose that T ( ) (cid:54) = ∅ , and the maximum of thesedimensions is p < n −
1. According to Theorem 2.19, there is an exponential set T ( ) such that T ( ) ⊂ T ( ) ⊂ T and dim T ( ) = p . If follows that T = T ( ) ∪ T ( ) ,hence, T is reducible which is a contradiction. Therefore, T ( ) = ∅ and T = T ( ) .Since T is irreducible, the set T ( ) consists of a unique hyperplane. (cid:3) Recall the definition of the ring K [ X , e X k ] , for 0 ≤ k ≤ n , in Remark 2.20. AssumingSchanuel’s conjecture, the following statement is a generalization of Corollary 4.7. Corollary 5.3.
Let P ∈ K [ X , U k ] and assume that Zer ( P ) ∈ R n + k is irreducible. Letf : = E ( P ) ∈ K [ X , e X k ] and Zer ( f ) be an ( n − ) -dimensional exponential set, irre-ducible in K [ X , e X k ] . Then, assuming Schanuel’s conjecture, Zer ( f ) is irreducible.Proof. Let Zer ( f ) be reducible. By Theorem 1.1, all ( n − ) -dimensional irreduciblecomponents of Zer ( f ) are rational hyperplanes, while by Theorem 2.19 and Re-mark 2.20, the union of all the rest of irreducible components is an exponential setdefined in K [ X , e X k ] . Since any rational hyperplane is an irreducible set defined in K [ X , e X k ] , we conclude that Zer ( f ) is also reducible in K [ X , e X k ] . (cid:3)
6. O
PEN QUESTIONS
1. Let dim ( Zer ( f )) = m ≤ n −
1, the algebraic set Zer ( P ) ⊂ R n be irreducible,and dim ( Zer ( P )) = k . Obviously, m ≤ k . Assume that k ≤ m + n . (In the caseof a single exponential, in Section 3, the condition k ≤ m + ( f ) is irreducible or every m -dimensional irreducible componentof Zer ( f ) is contained in a rational hyperplane through the origin. RREDUCIBLE COMPONENTS 15
2. We conjecture that there is an upper bound ( cd ) n on the number of all irre-ducible components of Zer ( f ) , similar to the bound in Corollary 3.8. Here c is anabsolute constant and d = deg ( P ) .3. It would be interesting to understand the structure of absolutely irreduciblecomponents of Zer ( f ) (i.e., irreducible over R ). Example 1.2 shows that Theo-rem 1.1 is no more true in this case.4. Let P ∈ K [ X , U ] , z ∈ Zer ( P , U − e X ) \ Zer ( X ) , and U be a neighbourhoodof z in R n + . Set Zer ( P ) ∩ U admits a Whitney stratification, while Zer ( U − e x ) is a real analytic submanifold of R n + .In Appendix, Theorem 7.2, it is proved that the intersection of Whitney stratifiedsets, Zer ( P ) ∩ U and Zer ( U − e X ) ∩ U , is transverse. We conjecture that transver-sality remains true in the general case of many exponentials.7. A PPENDIX
In this section we prove a transversality property for E -polynomials depending ona single exponential.The following statement is well known to experts but we could not find an exactreference to it in literature. Proposition 7.1.
Let
X ⊂ R n be an intersection of an algebraic set and an open set. Thenthere is a Whitney stratification of X (with connected strata) such that for each stratum Sthere is an open set U containing S such that S coincides with the intersection of U withan algebraic set. Moreover, if the algebraic set in X is defined over R alg then the algebraicset in S is defined over R alg .Proof. Let (cid:98)
X ⊂ C n be the complexification of X (i.e., the Zariski closure of X in C n ). Teissier’s theorem [15, Ch. VI, Proposition 3.1] implies that (cid:98) X admits aWhitney stratification, with each stratum being a Zariski locally closed set in C n .By taking connected components of real parts of strata, this stratification inducesthe required Whitney stratification on X .To prove the second statement of the proposition, observe that, according to [12],the existence of the required Whitney stratification for a fixed X can be expressedby a formula of the first-order theory of real closed fields. Now the statementfollows from the transfer principle in real closed fields [3, Proposition 5.2.3]. (cid:3) Let P ∈ K [ X , U ] , f = E ( P ) , V = Zer ( f ) , and π : R n + → R n be the projectionmap along U . Theorem 7.2.
Assume that for some ≤ p ≤ dim ( V ) there is a point x = ( x , . . . x n ) ∈ V p with x (cid:54) = . Let z ∈ Zer ( P , U − e X ) be such that π ( z ) = x . Then we have dim z ( Zer ( P )) = p + .Proof. Let U be a neighbourhood of z in R n + . By Proposition 7.1 there existsa Whitney stratification of Zer ( P ) ∩ U . Let S be a stratum of this stratificationcontaining z , and let S be an intersection of an algebraic set S (cid:48) and an open set in R n + .Note that S and Zer ( U − e X ) are real analytic submanifolds of R n + .To begin, we prove the following claim. Claim:
The manifolds S and Zer ( U − e X ) cannot be tangent at z . This claim implies that if dim ( S ) >
0, then S and Zer ( U − e X ) are transverse at z in R n + .To verify this claim we proceed by induction on dim S . Since any algebraic point inZer ( U − e X ) will require x =
0, the base case of the induction, with dim ( S ) = ( S ) = n − k + ≤ k < n + V of z in R n + such that V ∩ S = V ∩
Zer ( P , · · · , P k ) ,where all P i are polynomials in I ( S (cid:48) ) , and the Jacobian ( k × ( n + )) -matrix of thesystem P = · · · = P k = k at z . Now, assume that S andZer ( U − e X ) intersect tangentially at z . Then all ( k + ) × ( k + ) -minors of theJacobian ( k + ) × ( n + ) -matrix ∂ ( U − e X , P , . . . , P k ) ∂ ( U , X , . . . , X n ) vanish at z . In particular, all of the following minors vanish:(7.1) ∂ ( U − e X , P , . . . , P k ) ∂ ( U , X , X i , . . . , X i k − ) for all subsets { i , . . . , i k − } ⊂ {
2, . . . , n } .Also, if k < n , all of the following minors vanish:(7.2) ∂ ( P , . . . , P k ) ∂ ( X i , . . . , X i k ) for all subsets { i , . . . , i k } ⊂ {
2, . . . , n } .Clearly, the determinant of (7.1) equals D ( U , X , . . . , X n ) = det ∂ ( P , . . . , P k ) ∂ ( X , X i , . . . , X i k − ) − e X det ∂ ( P , . . . , P k ) ∂ ( U , X i , . . . , X i k − ) .Define (cid:98) D by replacing e X by U in D . Then (cid:98) D ( z ) = D ( z ) .Observe that A ( U , X , . . . , X n ) : = det ∂ ( P , . . . , P k ) ∂ ( U , X i , . . . , X i k − ) (cid:54) = z for some subset { i , . . . , i k − } ⊂ {
2, . . . , n } . Indeed, otherwise for all subsets { i , . . . , i k − } the condition D ( U , X , . . . , X n ) = B ( U , X , . . . , X n ) : = det ∂ ( P , . . . , P k ) ∂ ( X , X i , . . . , X i k − ) = z . Hence, all k × k -minors for the system P = · · · = P k = z , takinginto the account that all minors (7.2) vanish at z when k < n . This contradicts thesupposition that the Jacobian matrix of the system has the maximal rank at z .We conclude that A ( U , X , . . . , X n ) (cid:54) = z for some subset { i , . . . , i k − } ⊂{
2, . . . , n } . Fix such a subset { i , . . . , i k − } ⊂ {
2, . . . , n } . Then we can consider P = · · · = P k = F = ( F , F i , . . . , F i k − ) from the vector spaceof variables X , X j , . . . , X j n − k to the vector space of variables U , X i , . . . , X i k − , with { j , . . . , j n − k } = {
2, . . . , n } \ { i , . . . , i k − } . RREDUCIBLE COMPONENTS 17
In particular, there is a differentiable function F ( X , X j , . . . , X j n − k ) = U , whosepartial derivative with respect to X in the neighbourhood of z is given, accordingto formulae for differentiating of implicit functions, by ∂ F ∂ X ( X , X j , . . . , X j n − k ) = − B ( U , X , . . . , X n ) A ( U , X , . . . , X n ) .Suppose that (cid:98) D vanishes identically in the neighbourhood of z in S . Then, in theneighbourhood, U = B ( U , X , . . . , X n ) A ( U , X , . . . , X n ) ,and therefore, ∂ F ∂ X ( X , X j , . . . , X j n − k ) = − F ( X , X j , . . . , X j n − k ) .Let G be the restriction of F to the straight line Zer ( X j − x j , . . . , X j n − k − x j n − k ) .Then G satisfies the differential equation dG / dX = − G , hence G ( X ) = e − X .Since Zer ( G ( X ) − U ) is a semialgebraic curve at z , we get a contradiction. There-fore, (cid:98) D does not vanish identically in the neighbourhood of z in S .It follows that dim z ( (cid:98) D ∩ S ) < n − k +
1. The set (cid:98) D ∩ S is either smooth at z , or z is its singular point. In the first case, T z ( (cid:98) D ∩ S ) ⊂ T z ( S ) , hence (cid:98) D ∩ S is tangent toZer ( U − e X ) at z , which is impossible by the inductive hypothesis. In the secondcase, z belongs to a stratum of a smooth stratification of (cid:98) D ∩ S , which has evensmaller dimension than dim z ( (cid:98) D ∩ S ) . This is again impossible by the inductivehypothesis. Therefore, S and Zer ( U − e X ) do not meet tangentially at z . Theclaim is proved.To finish the proof of the theorem, we can assume that S is transverse to Zer ( U − e X ) at z . Let R be any other stratum of the stratification such that S ⊂ R . SinceZer ( U − e X ) is an oriented hypersurface in R n , there are two points a , b ∈ S ondifferent sides of Zer ( U − e X ) . There is an open curve interval γ ⊂ R such that a , b ∈ γ . Then γ ∩ Zer ( U − e X ) (cid:54) = ∅ , thus Zer ( U − e X ) ∩ R (cid:54) = ∅ . Since, by [16],Whitney’s ( a ) -regularity implies Thom’s ( t ) -regularity, the manifolds Zer ( U − e X ) and R are transverse in a neighbourhood of z . But dim z ( Zer ( P , U − e X )) = p ,while dim ( Zer ( U − e X )) = n −
1. It follows that dim z ( Zer ( P )) = p + (cid:3) A CKNOWLEDGEMENTS
We would like to thank L. Birbrair, N. Dutertre, and D. Trotman for communicat-ing to us a proof of the first part of Proposition 7.1. Part of the research presentedin this article was carried out during visits of the second author to the Aalto Sci-ence Institute in 2015/16. We are very grateful for financial support for these visitsfrom the Aalto Visiting Fellows programme.R
EFERENCES[1] A. Baker.
Transcendental number theory . Cambridge University Press, 1990. 10[2] R. Benedetti and J.-J. Risler.
Real algebraic and semi-algebraic sets.
Hermann, 1990. 5, 6[3] J. Bochnak, M. Coste, and M.-F. Roy.
Real algebraic geometry , volume 36 of
Ergebnisse der Mathematikund ihrer Grenzgebiete . Springer, 2013. 3, 15 [4] A. Gabrielov and N. Vorobjov. Complexity of computations with Pfaffian and Noetherian func-tions. In
Normal forms, bifurcations and finiteness problems in differential equations , pages 211–250.Kluwer, 2004. 1[5] R. Hartshorne.
Algebraic geometry , volume 52 of
Graduate Texts in Mathematics . Springer, 2013. 4, 5[6] B. Kazarnovskii. Exponential analytic sets.
Funct. Anal. Appl. , 31:86–94, 1997. 1[7] A. G. Khovanskii.
Fewnomials , volume 88 of
Translations of Mathematical Monographs . Amer. Math.Soc., 1991. 1[8] J. Kirby. Variants of Schanuel’s conjecture.
Preprint , 2007. 12[9] S. Lang.
Introduction to transcendental numbers , Addison Wesley, Reading, MA, 1966. 12[10] A. Macintyre and A. J. Wilkie. On the decidability of the real exponential field. In
Kreiseliana: aboutand around Georg Kreisel , 441–467. Wellesley, MA: A K Peters, 1996. 1[11] J. Pila. Functional transcendence via o-minimality. In
O-Minimality and Diophantine Geometry: Lon-don Mathematical Society Lecture Note Series: 421 , 66–99. Cambridge University Press, 2015. 12[12] E. Rannou. The complexity of stratification computation.
Discrete and computational geometry ,19:47–78, 1998. 15[13] M.-F. Roy and N. Vorobjov. Finding irreducible components of some real transcendental varieties.
Computational complexity , 4(2):107–132, 1994. 8[14] M.-F. Roy and N. Vorobjov. The complexification and degree of a semi-algebraic set.
MathematischeZeitschrift , 239:131–142, 2002. 10[15] B. Teissier. Variétés polaires. II: Multiplicités polaires, sections planes et conditions de Whitney. InAlgebraic geometry, Proc. int. Conf., La Rábida/Spain 1981, Lect. Notes Math. 961, pages 314–491.Springer, 1982. 15[16] D. Trotman. A transversality property weaker than Whitney (a)-regularity.
Bull. Lond. Math. Soc. ,8:225–228, 1976. 17[17] M. Waldschmidt. Private communication. 14[18] B. Zilber. Exponential sums equations and the Schanuel conjecture.
J. London Math. Soc.
ALTO S CIENCE I NSTITUTE , A
ALTO U NIVERSITY , P.O. B OX ALTO , F
INLAND D EPARTMENT OF M ATHEMATICS AND S TATISTICS , F
ACULTY OF S CIENCE AND T ECHNOLOGY , U NI - VERSITY OF T ROMSØ , 9037 T
ROMSØ ,N ORWAY
E-mail address : [email protected] D EPARTMENT OF C OMPUTER S CIENCE , U
NIVERSITY OF B ATH , B
ATH
BA2 7AY, E
NGLAND , UK
E-mail address ::