Intersecting varieties with tori

Boris Zilber zilber@maths.ox.ac.uk

20 December 2001

This paper discusses a conjecture of diophantine type that I came to in connection with an attempt to understand the complex exponentiation and other classical functions in model-theoretic context [Z1], [Z2], [Z3]. Not very surprisingly it turned out that the same conjecture could answer questions crucial for other model-theoretic works, see [P] and also an unfinished work by J.Baldwin and K.Holland. In [Z2] and [Z3] we need only the version over complex numbers with the exponentiation function, and in this form it is easier to see the links to the classical Schanuel conjecture. Nevertheless the discussion makes sense over any field K where an exponentiation-like function is defined (e.g. in ℝ) and also for much broader class of functions, e.g. elliptic and even Abelian functions, for which the ’Schanuel conjecture’ can be stated. In this broader context there is an obvious generalisation of the Diophantine conjecture that can be seen also as a generalisation of both Mordell-Lang and Manin-Mumford conjectures:

Conjecture on intersection with subgroups in semi-abelian varieties. Let A be the K-point set of a semi-abelian variety over K and V ⊆ A an algebraic subvariety. Then there is a finite collection τ(V) of cosets of some proper algebraic subgroups of A such that, given any algebraic subgroup B ≤ A and a component S ⊆ W ∩ B of the intersection which is atypical in dimension, i.e.

dim S > dim W + dim B − dim A,

there is a C ∈ τ(W) such that S ⊆ C.

We discuss here mostly the case A = (K^{∗})^{n}, a cartesian power of the multiplicative group of the field (and call it CIT, the conjecture on intersection with tori), K is ℂ or ℝ, but most of the statements can be easily generalised. One of the surprising observations is the fact that the real version of CIT, due to the o-minimality, follows from the Schanuel conjecture for the real exponentiation. Thus the multiplicative version of real Mordell-Lang conjecture follows from Schanuel. Presumably this is true for Abelian functions on reals, and thus general Mordell-Lang for reals follows from the corresponding ’Schanuel’.

1 Conjecture on intersection with tori

Definition A variety T_{m,b} ⊆ (K^{∗})^{n} given by a set of equations of the form

y_{1}^{m_{1}} · . . . · y_{n}^{m_{n}} = b  (1)

with m_{1}, …, m_{n} ∈ ℤ is said to be a [shifted] torus. Obviosly, a torus is a coset of a unique algebraic subgroup T_{m} ⊆ (K^{∗})^{n} defined by equations (1) with b = 1. This subgroup is a torus which we will call a basic torus and the base of T_{m,b}. A torus T ⊆ (K^{∗})^{n} is called proper if T ≠ (K^{∗})^{n}. A variety in K^{n} given by a set of equations of the form

m_{1} · x_{1} + … + m_{n} · x_{n} = a  (2)

with m_{1}, …, m_{n} ∈ ℤ is said to be a ℚ-affine space. If a = 0 the space is said to be ℚ-linear and is a base of the affine variety (2).

Lemma 1.1 For any ℚ-linear space N ⊆ K^{n} exp N is a torus and

dim N = dim exp N.

Proof Proceeding by induction we may assume that N is determined by one linear equation of the form (2) with a = 0 and g.c.d. of m_{1}, …, m_{n} is 1. Then (1) with b = 1 holds for any ȳ ∈ exp N. Conversely assume ȳ satisfies (1). Take x̄ ∈ K^{n} such that exp x̄ = ȳ. Then m_{1}x_{1} + … + m_{n}x_{n} = 2kπi. Let l_{1}, …, l_{n} ∈ ℤ be such that

l_{1}m_{1} + … + l_{n}m_{n} = 1

and define x′_{i} = x_{i} − l_{i}2kπi. Then

m_{1}x′_{1} + … + m_{n}x′_{n} = 0 and exp⟨x′_{1}, …, x′_{n}⟩ = ⟨y_{1}, …, y_{n}⟩.

Thus ȳ ∈ exp N and the latter is given exactly by (1). We see also that codim N = codim exp N, where codim is the number of independent equations which determine the linear space N in and the torus exp N in K^{n}, correspondingly. Hence dim N = dim exp N.□

Definition Let W ⊆ K^{n} be an algebraic variety defined over ℚ, T ⊆ (K^{∗})^{n} a torus, S an irreducible component of the algebraic variety W ∩ T. If dim S > dim W +dim T −n, then S is said to be an atypical component of the intersection W ∩ T. Otherwise S is said to be typical.

Conjecture on intersection with tori (CIT) For any W ⊆ K^{n} algebraic variety defined over ℚ there is a finite collection

τ(W) = {T_{1}, …, T_{k}}

of proper basic tori in (K^{∗})^{n} such that for any proper basic torus T ⊆ (K^{∗})^{n} and any atypical component S of W ∩ T

S ⊆ T_{i} for some T_{i} ∈ τ(W).

Notice that CIT implies a stronger form conjecture, when W is defined over ℚ̃, the algebraic closure of ℚ. Indeed, such a W has finitely many conjugates W^{σ} (σ a field automorphism) and W^{#} = ⋃_{σ} W^{σ} is defined over ℚ. Applying CIT to W^{#} we get τ(W).

Theorem 1 (CIT with parameters) Let V (a) ⊆ K^{n} be an algebraic variety depending on parameters a ∈ K^{k}. Then, assuming CIT, there is a finite set τ_{k}(V) of basic proper tori, a natural number t = t_{k}(V) and c_{1}(a), …, c_{t}(a) ∈ (K^{∗})^{n} depending on V and a such that given a proper basic torus T ⊆ K^{n} and an atypical component S_{a} of V (a) ∩ T there is P ∈ τ(V) and i ≤ t for which S_{a} ⊆ P · c_{i}(a).

Proof First we prove the statement with τ_{k}(V, a)) and t = t_{k}(V, a) depending both on V and a. For this w.l.o.g. we may assume that

V (a) = {⟨x_{1}, …, x_{n}⟩ : ⟨x_{1}, …, x_{n+1}, …, x_{n+k}⟩ ∈ V & x_{n+1} = a_{1}, …, x_{n+k} = a_{k}},

V ⊆ K^{n+k} is a ℚ̃-definable irreducible variety and a a generic point in pr(V), where pr : K^{n+k} → K^{k} is the projection onto the last k coordinates and is not contained in a proper subtorus of (K^{∗})^{k}. Notice that dim V (a) = dim V − dim prV then. Let S_{a} be a connected component of the intersection V (a) ∩ T for a torus T. Then S_{a} is definable over the algebraic closure of ℚ(a) and can be represented as an equidimensional connected component of an algebraic variety of the form S(a) for some S ⊆ K^{n+k} irreducible ℚ̃-definable. Notice that dim S(a) = dim S − dim prS and, since a ∈ prS, a is generic in prV and S ⊆ V, we have dim prV = dim prS. Thus

dim V − dim V (a) = dim S − dim S(a).

It follows that

dim S(a) > dim V (a)+dim T −n iff dim S > dim V +dim T ×(K^{∗})^{k}−(n+k).

The righthandside of the above is saying that S is an atypical component of the intersection of V and torus T × (K^{∗})^{k}. By CIT there is a proper torus P ∈ τ(V) such that S ⊆ P. Thus S(a) ⊆ P (a). But

P (a) = {⟨x_{1}, …, x_{n}⟩ : ⟨x_{1}, …, x_{n+1}, …, x_{n+k}⟩ ∈ P & x_{n+1} = a_{1}, …, x_{n+k} = a_{k}}

is a coset of a subtorus P^{(k)} ⊆ (K^{∗})^{n} and the subtorus is proper since otherwise a belongs to a proper subtorus of (K^{∗})^{k}, which contradicts the assumptions, and thus the statement depending on a has been proved. To prove the full statement suppose towards a contradiction that for any finite set τ of proper subtori of K^{n} and every natural number t there is an a ∈ K^{k} such that the pair ⟨τ, t⟩ does not fit for ⟨τ(V, a), t(V, a)⟩. It is a routine exercise to see that for every pair ⟨τ, t⟩ the set A_{τ,t} of such a′s is definable in the field language. Using Tarski-Seidenberg quantifier-elimination theorem for K we may say that A_{τ,t} is constructible, i.e. is obtained from algebraic subvarieties of K^{k} by finite unions, intersections and complements. Consequently, using the fact that K is of infinite transcendence degree, one can find an a ∈ K^{k} which belongs to A_{τ,t} for all ⟨τ, t⟩. This contradicts the above proved statement.□

Corollary 1 Let V (a) ⊆ K^{n} be an algebraic variety depending on parameters a ∈ K^{k} and b ∈ K^{n}. Then, assuming CIT, there is a finite set τ_{k}(V) of basic proper tori, a natural number t = t_{k}(V) and c_{1}(a, b), …, c_{t}(a, b) ∈ (K^{∗})^{n} depending on V and a, b such that given a proper basic torus T ⊆ K^{n} and an atypical component S_{a,b} of V (a) ∩ T · b there is P ∈ τ(V) and i ≤ t for which S_{a,b} ⊆ P · c_{i}(a, b).

We will need the following technical generalization of the notion of an atypical component of an intersection.

Definition Let T, P ⊆ K^{n} be tori, V ⊆ K^{n} an algebraic variety. Assume an irreducible component S of the intersection V ∩ T is a subset of P. Then S is said to be an atypical component of V ∩ T with respect to P iff

dim S > dim(V ∩ P) + dim(T ∩ P) − dim P.

Proposition 1 Let V (a) ⊆ K^{n} be an algebraic variety, b ∈ K^{n}. Then there is a finite set π(V) of basic tori of (K^{∗})^{n} a number p = p(V) and elements c_{1}(a, b), …, c_{p}(a, b) ∈ K^{n} such that given a basic torus T ⊆ K^{n} for any connected atypical component S of V (a) ∩ T · b there is Q ∈ π(V) and c_{i}(a, b) for which S ⊆ Q · c_{i}(a, b) and S is typical in V (a) ∩ T · b with respect to Q · c_{i}(a, b).

Proof We shall prove a slightly generalized version of the statement: Let P ⊆ K^{n} be a torus, V (a) ⊆ P an algebraic variety. Then there is a finite set π_{P} (V) of basic tori of (K^{∗})^{n} a number p = p_{P} (V) and elements c_{1}(a, b), …, c_{p}(a, b) ∈ K^{n} such that given a basic torus T ⊆ P for any connected atypical component S of V (a) ∩ T · b there is Q ∈ π_{P} (V) and c_{i}(a, b) for which S ⊆ Q · c_{i}(a, b) and S is typical in V (a) ∩ T · b with respect to Q · c_{i}(a, b). For dim P = 1 the statement is trivially true. Consider the general case. By CIT relativized to P there is a finite set τ_{P} (V) of proper subtori of P such that for any S ⊆ V ∩ P ∩ T atypical with respect to P there is a Q ∈ τ_{P} (V) with S ⊆ Q. By induction on dim P we can use the finite sets π_{Q}(V ∩ Q) of subtori of Q. Put

π_{P} (V) = ⋃_{Q∈τ_{P} (V)} π_{Q}(V ∩ Q).

□

2 CIT versus Mordell-Lang and Manin-Mumford

We discuss now the more general form of CIT as stated in the introduction. So A is a semi-abelian variety, and we write the group operation in A in the additive way. We show that in this form CIT is a conjecture of Diophantine type and is stronger than both Mordell-Lang and Manin-Mumford conjectures (now proved, see [L]). For simplicity we formulate both conjectures in a slightly restricted form.

Proposition 2 (Manin-Mumford case) Let W ⊆ A be an algebraic variety which contains no coset of an algebraic subgroup of A, and Γ the subgroup of torsion points of A. Then CIT formally implies that W ∩ Γ is finite.

Proof Each point in Γ of order p belongs to the 0-dimensional subgroup B_{p} = {a ∈ A : pa = 0}. By assumtions dim W < n. Thus any point in W ∩ Γ is an atypical component of the intersection W ∩ B_{p} and thus is contained in one of P belonging to τ(W). Since any coset P is a semi-abelian variety again, by induction on dim A it follows that W ∩ Γ ∩ P is finite, hence W ∩ Γ is finite.□

Proposition 3 (Mordell-Lang case) Let W ⊆ A be an algebraic subvariety which contains no coset of an algebraic subgroup of A, Γ a finitely generated subgroup of A. Then CIT formally implies that W ∩ Γ is finite.

Proof W.l.o.g. we assume W is not contained in any coset of a proper algebraic subgroup of A. We prove the statement by induction on dim A = n. For n = 1 the statement is trivial. Denote r the rank of group Γ (the size of a maximal independent subset) and dim W = w. Let k be an integer such that k(n − w) > nr, which exists since w < n. Consider any point ⟨a_{1}, …, a_{k}⟩ ∈ Γ^{k} ∩W^{k}. A maximal independent subset of a_{1}, …, a_{k} contains d ≤ r elements, say a_{1}, …, a_{d}, thus ⟨a_{1}, …, a_{k}⟩ belongs to a subgroup B_{a} ⊆ A^{k} defined by k − d equations of the form

mx_{d+i} = m_{i,1}x_{1} + … + m_{i,d}x_{d}, (i = 1, …, k − d)

for m and m_{i,j} integers. The equations are independent, hence dim B_{a} = nd and so ⟨a_{1}, …, a_{k}⟩ belongs to an atypical component of the intersection W^{k} ∩ B_{a} since

dim W^{k} + dim B_{a} − nk = wk + nd − nk ≤ nr − k(n − w) < 0.

By Theorem 1 there is a finite set of proper subgroups B_{1}, …, B_{l} of A^{k} and points c_{1}, …, c_{l} ∈ A^{k} such that (*) any point of the intersection W^{k} ∩ Γ^{k} belongs to one of the cosets B_{i} + c_{i} (i = 1, …, l). Suppose towards a contradiction that W ∩ Γ is infinite. Let now k be the minimal one with the property (*). Then there are only finitely many k-tuples w̄ ∈ W^{k} ∩ Γ^{k} with w_{i} = w_{j} for some 1 ≤ i < j ≤ k. Denote (W ∩Γ)^{(k)} the set of all k tuples with distinct coordinates from W ∩Γ. For any U ⊆ A^{k} denote

U^{sym} = {ū^{π} : ū ∈ U, π ∈ Sym(k) a permutation of coordinates}.

Then (B_{i} + c_{i})^{sym} is a finite union of cosets, and (W ∩ Γ)^{(k)} is a subset of the union of all (B_{i} + c_{i})^{sym}. By the classical combinatorial fact (Ramsey Theorem) there is an infinite X ⊆ W ∩ Γ and some i ∈ {1, …, l} such that

(W ∩ Γ)^{k} ∩ (B_{i} + c_{i})^{sym}.

It follows that there are a_{1}, …, a_{k−1} ∈ W ∩ Γ such that ⟨a_{1}, …, a_{k−1}, b⟩ ∈ (W ∩ Γ)^{k} ∩ (B_{i} + c_{i}) for infinitely many b ∈ W ∩ Γ.

But the condition on b

⟨a_{1}, …, a_{k−1}, b⟩ ∈ B_{i} + c_{i}

is equivalent to b ∈ B +d for some proper subgroup B ⊆ A and d ∈ Γ. Shifting W by d we may assume W ∩ Γ ∩ B is infinite. By induction hypothesis on n the only possibility then is W ∩ B is not proper in B, i.e. B ⊆ W. The contradiction.□

3 Schanuel Conjecture

We want to exhibit now some ties between CIT and the classical Schanuel Conjecture (SchC) For any ℚ-linearly independent complex numbers x_{1}, …, x_{n}

tr.d.(x_{1}, …, x_{n}, exp(x_{1}), …, exp(x_{n})) ≥ n.

It easy to eliminate the assumptions on the linear independence and see that the conjecture can be stated equivalently in the following form: For any x_{1}, …, x_{n} complex numbers

tr.d.(x_{1}, …, x_{n}, exp(x_{1}), …, exp(x_{n})) ≥ l.d.(x_{1}, …, x_{n}),

where l.d.(x_{1}, …, x_{n}) denotes the dimension of the ℚ-linear space generated by x_{1}, …, x_{n}.

Proposition 4 [Schanuel Conjecture with parameters] SchC implies: Let V ⊆ K^{2n+k} be an algebraic variety over ℚ̃, a = ⟨a_{1}, …, a_{k}⟩ ∈ K^{k} and

V (a) = {⟨x_{1}, …, x_{n}, y_{1}, …, y_{n}⟩ ∈ K^{2n} : ⟨x_{1}, …, x_{n}, y_{1}, …, y_{n}, a_{1}, …, a_{k}⟩ ∈ V }.

Suppose dim V (a) = d < n. Then there is a finite A ⊆ K, |A| ≤ k(n + 1), such that either x_{1}, …, x_{n} ∈ K^{n} are ℚ-linearly dependent over A or ⟨x_{1}, …, x_{n}, exp(x_{1}), …, exp(x_{n})⟩ ∉ V (a).

Proof If there are less than k + 1 n-tuples ⟨x_{1}, …, x_{n}⟩ such that ⟨x_{1}, …, x_{n}, exp(x_{1}), …, exp(x_{n})⟩ ∈ V (a) (call such n-tuples V (a)-tuples) then A is just the union of all elements in the tuples. So we assume one can choose k + 1 of them

⟨x_{i,1}, …, x_{i,n}, exp(x_{i,1}), …, exp(x_{i,n})⟩ ∈ V (a), i = 1, …, k + 1.

From the assumptions

tr.d.({x_{i,j}, a_{l}, exp(x_{i,j}), exp(a_{l}) : i ≤ k + 1, j ≤ n, l ≤ k}) ≤ (k + 1)d + 2k

and (k + 1)d + 2k < (k + 1)n + k. By SchC then

{x_{i,j} : i ≤ k + 1, j ≤ n} ∪ {a_{j} : j ≤ k}

are linearly dependent. Thus we have proved that for any k + 1 V (a)-tuples there is a linear dependence on their coordinates over a. Let now l ≤ k be maximal such that there is B = {b_{i,j} : i ≤ l, j ≤ n} independent over a and ⟨b_{i,1}, …, b_{i,n}⟩ are V (a)-tuples. Then A = B ∪ {a_{j} : j ≤ k} satisfies the requirement of the proposition.□

Proposition 5 SchC + CIT imply the following Uniform Schanuel Conjecture (USC): for any algebraic variety V ⊆ K^{2n} of dimension less than n there is a finite set μ(V) of proper ℚ-linear subspaces of K^{n} such that given

⟨x_{1}, …, x_{n}, exp(x_{1}), …, exp(x_{n})⟩ ∈ V

there is M ∈ μ(V) and an integer vector z̄ ∈ ℤ^{n} such that

⟨x_{1}, …, x_{n}⟩ ∈ M + 2πi · z̄

and M is of codimension at least 2 or z̄ = 0.

Remark We can equivalently state that for any connected component S of V ∩ {exp(x_{i}) = y_{i}, i = 1, … n} there is M and z̄ as above such that pr_{f}S ⊆ M + 2πi · z̄ (projection onto the first n coordinates). Proof We assume SchC+CIT and prove USC. Denote pr_{l} the projection K^{2n} → K^{n} onto the last n coordinates. Let

dim(V ∩ pr_{l}^{−1}(y)) = d

for a generic y ∈ pr_{l}(V). Then dim V = dim pr_{l}(V) + d. Under our asumptions, by SchC there is a proper ℚ-linear subspace N of K^{n} which contains ⟨x_{1}, …, x_{n}⟩, more precisely, dim N = l.d.(x_{1}, …, x_{n}). Hence ⟨exp(x_{1}), …, exp(x_{n})⟩ is an element of the proper subtorus T = exp(N) of (K^{∗})^{n}. Denote algebraic varieties

V′_{l} = {y ∈ V_{l} : dim(V ∩ pr_{l}^{−1}(y)) > d}, V′ = {⟨x, y⟩ ∈ V : y ∈ V′_{l}}.

Since V′ is a proper algebraic subvariety of the irreducible variety V,

dim V′ < dim V.

Denote also

d′ = min{dim(V ∩ pr_{l}^{−1}(y)) : y ∈ V′_{l}}.

Consider two cases. Case 1: ⟨x_{1}, …, x_{n}, exp(x_{1}), …, exp(x_{n})⟩ ∈ V′. Then

tr.d.(x_{1}, …, x_{n}, exp(x_{1}), …, exp(x_{n})) ≤ d′+dim(V′_{l}∩T) ≤ dim V′+dim T −n+s′ < dim T +s′,

where s′ = dim V′_{l} + dim T − n − dim(V′_{l} ∩ T). By SchC s′ > 0, which means the intersection V′_{l} ∩ T is atypical. Hence ⟨exp(x_{1}), …, exp(x_{n})⟩ ∈ P_{i} for one of tori P_{i} ∈ τ(V′_{l}). Case 2: ⟨x_{1}, …, x_{n}, exp(x_{1}), …, exp(x_{n})⟩ ∈ V \ V′. Then

tr.d.(x_{1}, …, x_{n}, exp(x_{1}), …, exp(x_{n})) ≤ d+dim(V_{l}∩T) ≤ dim V +dim T −n+s < dim T +s, where s = dim V_{l} + dim T − n − dim(V_{l} ∩ T). By SchC s > 0, which means the intersection V_{l} ∩ T is atypical. Hence ⟨exp(x_{1}), …, exp(x_{n})⟩ ∈ Q for one of tori Q ∈ π(V_{l}). Denote now μ(V) the set of ℚ-linear subspaces M of K^{n}, such that exp(M) = P for P ∈ π(V_{l}) ∪ π(V′_{l}.) Suppose now y = ⟨exp(x_{1}), …, exp(x_{n})⟩ ∈ P, exp(M) = P for P ∈ π(V_{l}) and codim P = 1. Denote V_{2} = pr_{l}(V). Then by definition of π(V_{l})

dim_{y}(V_{2} ∩ T ∩ P) = dim(T ∩ P) + dim_{y}(V_{2} ∩ P) − dim P

(dim_{y} is the dimension of a maximal component containing y.) Since dim(T ∩ P) = dim T + dim P − n + s_{T,P} for some s_{T,P} ≥ 0 and dim_{y}(V_{2} ∩ P) = dim P + dim V_{2} − n + s_{V,P} for some s_{V,P} ≥ 0, we get

dim_{y}(V_{2} ∩ T ∩ P) = dim T + dim P + dim V_{2} − 2n + s_{T,P} + s_{V,P} .

On the other hand

dim_{y}(V_{2}∩T ∩P) = dim_{y}(V_{2}∩T)+dim P −n = dim V_{2}+dim T +s+dim P −2n

for s > 0 defined above. Thus s = s_{T,P} + s_{V,P} . If s_{T,P} = 0, then s_{V,P} > 0, which implies V_{2} ⊆ P and

dim_{y}(V_{2} ∩T ∩P) = dim(T ∩P)+dim V_{2} −dim P = dim T +dim V_{2} −n+s_{T,P} .

Since

dim_{y}(V_{2} ∩ T ∩ P) = dim_{y}(V_{2} ∩ T) = dim T + dim V_{2} − n + s_{V,P} ,

we get s_{V,P} = s_{T,P} , a contradiction. Thus T and P intersect atypically (i.e. not transversally), which means T ⊆ P in case codim P = 1. It follows N ⊆ M and hence ⟨x_{1}, …, x_{n}⟩ ∈ M in this case. Otherwise ⟨x_{1}, …, x_{n}⟩ ∈ M + 2πi · z̄ for some integer vector z̄. In case y ∈ pr_{l}(V′) the same arguments applied to V′ again give the required.□

Lemma 3.1 USC implies SchC.

Proof Notice that

tr.d.(x_{1}, …, x_{n}, exp(x_{1}), …, exp(x_{n})) < n

is equivalent to

⟨x_{1}, …, x_{n}, exp(x_{1}), …, exp(x_{n})⟩ ∈ V

for some ℚ-definable algebraic variety V of dimension less than n. By USC either x_{1}, …, x_{n} are ℤ-linearly dependent, or there are two independent ℤvectors ⟨z_{j,1}, …, z_{j,n}⟩ and k_{j} ∈ ℤ (j = 1, 2) such that

z_{j,1} · x_{1} + … + z_{j,n} · x_{n} = 2k_{j}πi (j = 1, 2).

It follows there is a nonzero integer vector ⟨z_{1}, …, z_{n}⟩ such that

z_{1} · x_{1} + … + z_{n} · x_{n} = 0,

which is the statement of SchC.□

Lemma 3.2 USC implies: Let W (ā) ⊆ K^{n} be an algebraic variety, Then there is a finite set μ(W) of proper ℚ-linear subspaces of K^{n} and a number m = m(W) such that, given L ⊆ K^{n} a K-linear subspace and R ⊆ L ∩ ln W (ā) a connected component of the analytic subset of K^{n}, there are c̄_{1}, …, c̄_{m} ∈ K^{n} satisfying R ⊆ M + c_{i} + 2πiz̄ for some i ≤ m, M ∈ μ(W) and z̄ ∈ ℤ^{n}.

Proof By assumptions dim R > dim L + dim W (ā) − n. By intersecting R and L with generic hyperplanes we can reduce the dimensions of R and L, if needed, and come to the situation dim R > dim L + dim W (ā) − n < 0, which we will further assume. Let L be given by matrix equation Bx̄ = 0. Consider the algebraic variety

V (ā, B) = {x̄ ∈ K^{n} : Bx̄ = 0} × W (ā).

By the assumtion dim V (ā, B) < n and S = {⟨x̄, exp(x̄)⟩ : x̄ ∈ R} satisfies the assumptions of Proposition 5 (the remark) with parameters. Hence the statement follows.□

Corollary 2 USC implies CIT with parameters.

Repeating the arguments in the proof of Proposition 1 we get from Lemma 3.1

Corollary 3 USC implies: Let W (ā) ⊆ K^{n} be an algebraic variety, Then there is a finite set λ(W) of proper ℚ-linear subspaces of K^{n} and a number l = l(W) such that, given L ⊆ K^{n} a K-linear subspace and R ⊆ L ∩ ln W (ā) a connected component of the analytic subset of K^{n}, there are c̄_{1}, …, c̄_{l} ∈ K^{n} satisfying R ⊆ M + c_{i} + 2πiz̄ for some i ≤ m, M ∈ λ(W) and z̄ ∈ ℤ^{n}, and R is typical in L ∩ ln W (ā) with respect to M + c_{i} + 2πiz̄.

Lemma 3.3 In Corollary above, denoting M′ = M + c_{i} + 2πiz̄,

dim(M′ ∩ L) > dim M′ + dim L − n

and

dim(M′ ∩ ln W (ā)) > dim M′ + dim W (ā) − n.

Proof By the assumption that R is typical with respect to M′

dim R = dim M′ ∩ L + dim_{R} M′ ∩ ln W (ā) − dim M′,  (3)

where dim_{R} M′ ∩ ln W (ā) is the dimension of the connected component of the intersection containing R. Let

dim M′ ∩ L = dim M′ + dim L − n + d_{L},  (4)
dim_{R} M′ ∩ ln W (ā) = dim M′ + dim W (ā) − n + d_{W} ,  (5)

where d_{L}, d_{W} are some non-negative integers which are just zero iff the corresponding intersections are transversal (intersect typically). It follows from (3) and (4) that

dim R = dim L − n + d_{L} + dim_{R} M′ ∩ ln W (ā) ≤ dim L + dim W (ā) − n + d_{L}

but dim R > dim L + dim W (ā) − n by the assumptions. It follows d_{L} ≥ 1, hence M′ intersects L not transversally. Symmetrically, using (3) and (5), d_{W} ≥ 1.□

Theorem 2 Assume USC. Then given a K-linear L ⊆ K^{n} not contained in any proper ℚ-linear subspace of K^{n}, and an algebraic family W (ā) ⊆ K^{n} of algebraic varieties there is a finite set η(L, W) of K-linear hyperplanes of L and a natural number l = l(W) such that there are d_{1}, …, d_{l} ∈ K^{n} satisfying the property that any atypical component R of L ∩ ln W (ā) is contained in some H + d_{i} for H ∈ η(L, W) and i ≤ l.

Proof By Corollary 3 the atypical component R is contained in some M′ = M +c+2πiz̄, c taking one of the l(W) values depending on ā, M a ℚ-linear subspace from ν(W) and R is typical with respect to M′. Since M′ intersects with L we may assume that c ∈ L. By ℚ-linear transformation of variables we may assume that M is given by p = codim M equations

x_{1} = … = x_{p} = 0,

and M + c + 2πiz̄ is given by

x_{1} = 2\frac{k_{1}}{m}πi + c_{1}, … x_{l} = 2\frac{k_{l}}{m}πi + c_{p}

for k_{1}, …, k_{p}, m ∈ ℤ, c_{1}, …, c_{p} ∈ K. By Lemma 3.3 M is not transversal to L, hence there is a non-trivial K-linear equation of the form s_{1}x_{1}+…+s_{p}x_{p} = 0 which holds on L. In other words the projection of L onto x_{1}, …, x_{p} denoted prL is a proper subspace of K^{p}. Let N be the minimal K-linear subspace of C^{p} such that

ℤ^{p} ∩ N = ℤ^{p} ∩ prL.

In fact, N is ℚ-linear since it has a basis consisting of integer vectors. Since L is not contained in a proper ℚ-linear subspace, N ∩ prL is a proper subspace of prL. Let ⟨x_{1}, …, x_{n}⟩ ∈ R. Then by the construction ⟨x_{1}, …, x_{p}⟩ ∈ prL and ⟨x_{1}, …, x_{p}⟩ = 2πi\frac{1}{m} z̄ + pr(c) for some integer vector z̄. Since pr(c) ∈ prL, ⟨x_{1}, …, x_{p}⟩ ∈ \frac{1}{m} · ℤ^{p} ∩ prL ⊆ N. Denote H = \{y ∈ L : pr(y) ∈ N\}, which by the construction is a proper K-linear subspace of L and R ⊆ H + c. Notice that we constructed H using only M and L and c depends only on the coset M′. Define η(L, W) to be all the H constructed from M of ν(W) and l(W) the same as in Corollary 3.□

4 The ’function field’ case of CIT

During 1997-98 Logic Year in MSRI, Berkeley, E.Hrushovski pointed out that CIT can be treated by means of the theory of differential fields, and C.Wood and A.Pillay showed me the paper by J.Ax which contains the following basic theorem Theorem of J.Ax [A] Let y_{1}, …, y_{n}, z_{1}, …, z_{n} be non-zero elements of a differential field with subfield of constants C ⊃ ℚ such that Dy_{i} = \frac{Dz_{i}}{z_{i}} for i = 1, …, n, and Dy_{1}, …, Dy_{n} are ℚ-linearly independent. Then

tr.d._{C}(y_{1}, …, y_{n}, z_{1}, …, z_{n}) ≥ n + 1.

In the context of the broader conjectures it is interesting to mention that W.D.Brownawell and K.K.Kubota in [BK] generalized the Ax’s result to Weierstrass elliptic functions.

Proposition 6 Given V ⊆ K^{2n+k} an algebraic variety there is a finite collection U of non-zero integer vectors such that for any ā ∈ K^{k} with dim V (ā) ≤ n for any infinite component S ⊆ V (ā) ∩ {exp(x_{i}) = x_{n+i}, i = 1, … n} for one of u ∈ U there is a constant c_{S} such that all ⟨s_{1}, …, s_{2n}⟩ ∈ S satisfy u_{1}s_{1} + … + u_{n}s_{n} = c_{S}.

Proof Take y_{1}, … y_{n}, z_{1}, …, z_{n} to be the coordinate functions on V (ā). Consider now the restriction of functions y_{1}, …, y_{n}, z_{1} …, z_{n} to S and differentiation D = \frac{d}{dt} along a curve in a simple point s of the component. Then

Dz_{i} = z_{i}Dy_{i} for i = 1, …, n  (6)

in the differential field F_{S,s} of germs of analytic functions on S in s, and

⟨y_{1}, … y_{n}, z_{1}, …, z_{n}⟩ ∈ V (ā)(F_{S,s}).  (7)

Now assume towards a contradiction that for any choice of finite collection U of integer vectors there is ā and an infinite component Sā ⊆ V (ā) such that the statement of the proposition fails. Then in some formal differential field F (6) and (7) hold with F in place of F_{S,s}, and Dy_{1}, …, Dy_{n} are ℚ-linearly independent. This contradicts the theorem of Ax.□

It follows from the results of the preceding section

Corollary 4 Given an algebraic variety W (ā) ⊆ K^{n} there is a finite collection μ(W) of non-zero integer vectors such that for any torus T ⊆ K^{n} and an infinite atypical component S ⊆ W (ā) ∩ T of the intersection there is m̄ ∈ μ(W) and a constant c such that all (a_{1}, …, a_{n}) in the component satisfy a_{1}^{m_{1}} · . . . · a_{n}^{m_{n}} = c.

Correspondingly the proof of Theorem 2 yields

Corollary 5 Given a K-linear L ⊆ K^{n} not contained in any proper ℚ-linear subspace of K^{n}, and an algebraic family W (a) ⊆ K^{n} of algebraic varieties there is a finite set η(L, W) of K-linear hyperplanes of L such that any atypical infinite component R of L ∩ ln W (a) is contained in some H + d for H ∈ η(L, W) and d ≤ L.

5 The real exponent, real CIT and the Mordell conjecture

Theorem 3 If the real exponentiation satisfies the Schanuel conjecture then it satisfies the Uniform Schanuel conjecture.

Proof ^{1} Let V ⊆ ℝ^{2n} be an algebraic variety, dim V < n. Denote

V^{e} = {⟨x_{1}, …, x_{n}, exp(x_{1}), …, exp(x_{n})⟩ ∈ V }.

Then by o-minimality of ℝ_{exp} (see [W], but in fact the Hovanski’s results [Ho] suffice) V^{e} is an analytic subspace with finitely many connected components V_{i}^{e}, i = 1, …, k. By the Schanuel conjecture

V^{e} ⊆ ⋃_{M⊂ℝ^{n} ℚ-linear subspace}{⟨x_{1}, …, x_{n}, exp(x_{1}), …, exp(x_{n})⟩ : {⟨x_{1}, …, x_{n}⟩ ∈ M}

1This proof is incomplete. For the full proof see the joint note with J.Kirby The uniform Schanuel conjecture over the real numbers. The author’s web-page and to appear in the Bulletin of LMS

thus, by the connectedness, each component V_{i}^{e} lies in a set of the form {⟨x_{1}, …, x_{n}, exp(x_{1}), …, exp(x_{n})⟩ : {⟨x_{1}, …, x_{n}⟩ ∈ M}.□

Corollary 4 for reals by the same o-minimality argument takes the form

Corollary 6 ^{2} Given an algebraic variety W (ā) ⊆ ℝ^{n} there are a finite collection μ(W) of non-zero integer vectors, a number m(W) independent on a, and c_{1}, …, c_{m(W)} ∈ ℝ^{n}, depending on a, such that for any torus T ⊆ ℝ^{n} and an infinite atypical component S ⊆ W (ā) ∩ T of the intersection there is m̄ ∈ μ(W) and i ≤ m(W) such that all ⟨a_{1}, …, a_{n}⟩ in the component satisfy a_{1}^{m_{1}} · . . . · a_{n}^{m_{n}} = c_{i}.

References

[A] J.Ax, On Schanuel’s conjectures, Annals of Mathematics, 93 (1973), 252-258

[BK] W.D.Brownawell, K.K.Kubota, The algebraic independence of Weierstrass functions and some related numbers, Acta Arithmetica, 32 (1977), 111-149

[L] S.Lang, Number Theory III, Encyclopaedia of Mathematical Sciences, Springer-Verlag, 1991

[P] B.Poizat, L’egalite au cube, Prepublication, 2000

[W] A.Wilkie, On the theory of the real exponential field, Illinois J. of Math., 33 (1989) no 3

[Z1] B.Zilber, Fields with pseudo-exponentiation, Prepublication, 1999 www.maths.ox.ac.uk/z̃ilber

[Z2] B.Zilber, Raising to powers in algebraically closed fields, Prepublication, 1999 www.maths.ox.ac.uk/z̃ilber [Z3] B.Zilber, On the systems of exponential sums with real exponents, Prepublication, 2000 www.maths.ox.ac.uk/z̃ilber

2Ignore this. The o-minimality argument does not work here. February, 2006