On systems of exponential sums with real exponents

Boris Zilber zilber@maths.ox.ac.uk

June 6, 2000

In [Z2] we studied the theory of formal exponentiation (raising to powers) and proved that it is very nice (superstable) provided a certain diophantine conjecture CIT is true (see [Z1] for a discussion of CIT). We believe that in fact the theory describes the genuin complex exponentiation. To show this we need two statements, one of them being the Schanuel conjecture [L] and the other is the exponential-algebraic closedness of C, i.e. the solvability of any non-obviously-inconsistent system of equations. Here we give a proof of the latter in case of real powers modulo the Schanuel conjecture and CIT. The proof is based on the theory of exponential sums developed by D.Bernstein, A.Kushnirenko, B.Kazarnovski and A.Khovanski, see [Kh] (the Russian edition).

1 Exponential sums

Let V ⊆ C^{n} be a zero-set of p polynomials

f_{i}(y) = ∑_{m∈Γ_{i}} a_{i,m}y^{m} (i = 1, …, p)  (1)

where y = ⟨y_{1}, …, y_{n}⟩ are variables, m = ⟨m_{1}, …, m_{n}⟩ ∈ Z^{n}, y^{m} = y_{1}^{m_{1}} · . . . y_{n}^{m_{n}}, a_{m} ∈ C and Γ_{i} some finite subsets of Z^{n}. Let L ⊆ C^{n} be a K-linear subspace of C^{n} for some subfield K ⊆ C. Up to the numeration of coordinate functions x_{1}, …, x_{n} we may assume L is given by n − k linear equations of the form x_{j} = (s_{j}, z), j = k+1, …, n, where z = ⟨x_{1}, …, x_{k}⟩, s_{j} = ⟨s_{j,1}, …, s_{j,k}⟩ ∈ K^{k} and (s_{j}, z) = ∑_{l≤k} s_{j,l}x_{l}. We assume also s_{j} = ⟨0, …, 0, 1, … 0⟩ with 1 on jth place for j = 1, …, k, thus

(s_{j}, z) = x_{j} for all j = 1, …, n.  (2)

We also identify the points t ∈ K^{n} with the K-linear mapping (t, x) = t_{1} · x_{1} + … + t_{n} · x_{n} from C^{n} to C. The restriction of such a mapping to L denote t_{L}. Denote λ the linear mapping t → t_{L} from the space K^{n} of K-linear forms on C^{n} onto the space of K-linear forms L^{∗} on L. With this notation we associate to the system (1) the system of exponential sums in coordinates z ranging in L

∑_{m∈Γ_{i}} a_{i,m} exp(m_{L}, z) (i = 1, …, p)  (3)

or, equivalently, after gathering summands with common exponent, we get from (1) the system of equations

x ∈ L & ⋀_{i=1,...,p} ∑_{d∈Δ_{i}} b_{i,d} exp(d, x) = 0  (4)

with Δ_{i} = {m_{L} : m ∈ Γ_{i}} ⊆ L^{∗} and

b_{i,d} = ∑_{{m∈Γ_{i}: d=λ(m)}} a_{i,m}.

The set of solutions of the system (4) can be identified as L ∩ ln V, where

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

Thus the correspondence between pairs (L, V) and exponential systems, depending only on the choice of coordinate functions, is established.

Definition Let L be a linear subspace of C^{n} and V an algebraic variety in C^{n}. Given a pair (L, V), a Q-affine subspace M of C^{n} and a point a ∈ M ∩ L ∩ ln V it is said that (L, V) restricted to M in a is free if M is the minimal Q-affine subspace of C^{n} containing M ∩ L and exp(M) is the minimal (shifted) torus containing the connected component of V ∩ exp(M) passing through exp(a). If M = C^{n} in the definition and exp(C^{n}) is the minimal torus containing a connected component of V then (L, V) is said to be free.

Notice that if L is free of additive dependencies, in particular, if the pair if free, then for any d ∈ K^{k} there is no more than one m ∈ Z^{n} such that m_{L} = d, in other words the mapping t ↦ t_{L} is injective on Z^{n}. In this case also b_{i,d} = a_{i,m} for d = m_{L} in equations (4).

The following notions are basic in [Z].

Definition A pair (L, V) with V defined over A ⊆ C is said to be normal if there are ⟨a_{1}, …, a_{n}⟩ ∈ L and ⟨b_{1}, …, b_{n}⟩ ∈ V such that for any l ≤ n independent integer vectors m_{i} = ⟨m_{i,1}, … m_{i,n}⟩, i = 1, …, l, and

a′_{i} = m_{i,1}a_{1} + … + m_{i,n}a_{n}, b′_{i} = b_{1}^{m_{i,1}} · . . . · b_{n}^{m_{i,n}}

it holds

lin.d._{K}(a′_{1}, …, a′_{l}) + tr.d.(b′_{1}, …, b′_{l}/A) ≥ l.

Equivalently, for the K-linear subspaces L′_{1,...,l} = linlocus⟨a′_{1}, …, a′_{l}⟩, the minimal Klinear subspace of C^{l} containing ⟨a′_{1}, …, a′_{l}⟩, and varieties W′_{1,...,l} = alglocus_{A}⟨b′_{1}, …, b′_{l}⟩, the minimal algebraic subvariety of C^{l} defined over A and which contains ⟨b′_{1}, …, a′_{l}⟩, it must hold

dim L′_{1,...,l} + dim W′_{1,...,l} ≥ l.

The following connects the normality and freeness conditions to the basic invariant of the theory of exponential sums, the mixed Minkovski volume of Newton polytopes of the system, see [Kh] and [GW].

Lemma 1.1 Assume V is given by a system f = 0 of p = k independent polynomial equations. If (L, V) is normal and free and dim L + dim V = n, (dim L = k) then for the corresponding exponential system of k equations in k variables the mixed Minkovski volume of convex envelopes Δ̄_{1}, … Δ̄_{k}

Vol(Δ̄_{1}, … Δ̄_{k}) ≠ 0.

Proof By a suitable replacement of variables we may assume that in each equation non-trivial constant term occurs. I.e. each Γ_{i} contains the zero-point. We use the fact that, assuming each Δ̄_{i} contains the zero point, Vol(Δ̄_{1}, … Δ̄_{k}) ≠ 0 provided that for any distinct i_{1}, …, i_{l} ≤ k

lin.d.span_{K}(Δ_{i_{1}} ∪ … ∪ Δ_{i_{l}}) ≥ l.

Notice that, since V is free of multiplicative dependencies and V is non-empty, V ∩ (C^{∗})^{n} ≠ ∅. Notice that since L has no additive dependencies the mapping m ↦ m_{L} is injective on Z^{n} and (Γ_{i})_{L} = Δ_{i}. Suppose towards a contradiction that span_{K}(Δ_{1}, … Δ_{l}) = H_{Δ} ⊆ L^{∗} is of dimension less than l. Since the mapping λ : K^{n} → L^{∗} is linear, for H_{Γ} = span_{K}(Γ_{1}, … Γ_{l}),

λ(H_{Γ}) = H_{Δ}

and

dim ker_{Γ} λ = dim H_{Γ} − dim H_{Δ},

for ker_{Γ} λ, the kernel of the mapping restricted to H_{Γ}. Denote s = dim H_{Γ}. Then

dim ker_{Γ} λ > s − l.

Since H_{Γ} is generated by integer points, there is a Z-linear transformation of variables on C^{n} of the form x′_{i} = (q_{i}, x), for some q_{i} ∈ Q^{n}, such that q_{1}, …, q_{s} is a basis of H_{Γ} which also generates the abelian subgroup containing Γ_{1}, …, Γ_{l}. Letting y′_{i} = exp(x′_{i}) we can equivalently rewrite equations (1) so that in the first l equations only y′_{1}, …, y′_{s} occur. Denote V_{s} the projection of V onto the subspace of the first variables y′_{1}, …, y′_{s} and L_{s}, correspondingly, the projection of L onto (x′_{1}, …, x′_{s})-subspace C^{s}. Since the equations f = 0 were independent, dim V_{s} ≤ s − l. The dimension of L_{s} we deduce from that of ker_{Γ} λ. Notice that ker_{Γ} λ is the set of all K-linear t : C^{s} → C which are zero on L_{s}. Thus the quotient K^{s}/ ker_{Γ} λ can be identified with the space L_{s}^{∗} of all K-linear forms on L_{s}, the dual space to L_{s}. Since dim L_{s}^{∗} = dim L_{s}, we get

dim L_{s} = s − dim ker_{Γ} λ.

Thus

dim V_{s} + dim L_{s} ≤ (s − l) + (s − dim ker_{Γ} λ) < s,

which contradicts the normality of (L, V).□

We use below the widely known Schanuel Conjecture SchC and a conjecture of diopantine type, the conjecture on intersection varieties with tori in atypical components CIT, discussed in [Z]. It is proved in [Z] Theorem 1 Assume SchC+CIT. Then given a K-linear L ⊆ C^{n} not contained in any proper Q-linear subspace of C^{n}, and an algebraic family W (ā) ⊆ C^{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} ∈ C^{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.

Notation Fix an exponential system given by p = k polynomial equations (1) in C^{n} and a K-linear subspace L ⊆ C^{n} in the final form (4). Denote V the family of all systems V (a) of p = k independent equations (1). We assume all a_{i,m} ≠ 0 in the initial system and also that L is free, i.e. not contained in any proper Q-linear subspace of C^{n}. We also assume that the projective completion V̄ (a) ⊆ P^{n}(C) of V (a) has no component lying in the subvariety y_{i} = 0 or y_{i}^{−1} = 0 for i ≤ n. It is well known that the family V can be coordinatized by points of C^{d} for some d so, that the coordinatization identifies family V as an algebraic set C(V), i.e. a set of the form S_{1} \ S_{2} with S_{1} and S_{2} algebraic varieties, and each point of C(V) corresponds to an algebraic variety V (a) of the family. W.l.o.g. we assume C(V) is irreducible (i.e. S_{1} is).

We give now several key definitions and results from [Kh], see also earlier publication [K]. We assume from now on K = R ( [Kh] considers only this case).

Definition Let Σ_{i} ⊆ Γ_{i} be faces of corresponding convex polytopes. The faces are said to agree if there is a common R-linear function φ on R^{n} that takes its minimum on each convex polytope exactly on Σ_{i}. To each agreed set Σ_{1}, …, Σ_{k} of faces one associates a shortening of the initial system f = 0 of equations defined as

f_{i}^{Σ_{i}} = ∑_{m∈Σ_{i}} a_{m}y^{m} = 0 (i = 1, …, k).

If f = 0 determines variety V ⊆ C^{n} we denote V^{Σ} the variety corresponding to the shortened system.

Correspondingly the definition is applicable to any exponential system with real powers.

One can easily see that under assumption that an exponential system is associated to a pair (L, V) any shortening of the exponential system has the form L ∩ ln V^{Σ} for a correspondent shortening V^{Σ} of V.

Definition Given G, a domain in R^{k}, the exponential system associated to a pair (L, V) is said to be non-degenerate on infinity in domain R^{k} × ıG (ı^{2} = 1) if all points of L ∩ ln V lying in the domain are isolated and all shortenings L ∩ ln V^{Σ} do not have points in the domain.

Definition An ultrafilter D on N is a family of subsets of N satisfying for any X, Y ⊆ N the conditions:

(i) X, Y ∈ D implies X ∩ Y ∈ D; (ii) X ∈ D and X ⊆ Y implies Y ∈ D; (iii) X ∉ D iff N \ X ∈ D.

In a τ_{1}-topological space a point ξ is said to be the limit point of a sequence ζ = {ζ_{i} : i ∈ N} along an ultrafilter D if for any neighborhood V of ξ

{i ∈ N : ζ_{i} ∈ V } ∈ D.

Notation We denote the limit point ξ = ζ/D. Remark In a compact topological space ζ/D always exists.

Lemma 1.2 For L and V as above there is a finite set ρ(L, V) of algebraic families of algebraic varieties W such that for any W ∈ ρ(L, V) there are l = l(W) < n and a Q-linear transformation of basis of C^{n} to a new basis x_{1}, …, x_{n}, which induces the basis y_{1}, …, y_{n} (y_{i} = exp(x_{i})) of C^{n} such that

(i) the projection pr_{l}(L) of L into the (x_{1}, …, x_{l})-subspace S ⊆ C^{n} along (x_{l+1}, …, x_{n}) is of dimension less than dim L; (ii) for any a ∈ C(V) W (a) is a subvariety of the space exp(S) and l − dim W (a) > n − dim V; (iii) given sequences {a_{i} ∈ C(V) : i ∈ N}, ζ(i) ∈ exp(L) ∩ V (a_{i}) and an ultrafilter D on N, the limit point ζ/D of the sequence along D belongs to the projective closure V̄ (a) for some a. Suppose a ∈ C(V) (i.e. dim V (a) = n − k) but ζ/D ∉ (C^{∗})^{n}. Then there is W ∈ ρ(L, V) such that pr_{l}(ζ/D) ∈ W (a) ∩ exp pr_{l}(L).

Proof Let ζ = ⟨ζ_{1}, …, ζ_{n}⟩ be as in (iii). Then up to numeration of variables there is an l < n such that ζ_{l+1}/D, …, ζ_{n}/D takes values 0 or ∞ and ζ_{1}/D, …, ζ_{l}/D are in C^{∗}. By transformations of the form y_{i} → y_{i}^{−1} (corresponds to the sign-changing rational transformation x_{i} → −x_{i}) we may assume

ζ_{l+1}/D = … = ζ_{n}/D = 0.

Notice that there are only finitely many choices of subsets of variables and signchanging transformations for all choices of ζ. From that point on we begin an algorithm constructing a new basis. For any non-negative integer d such that d > dim V (a) + l − n for any distinct i_{1}, …, i_{d} ∈ {1, …, l} (if d = 0 then the set is empty) the projection pr_{i_{1},...,i_{d},l+1,...,n}V onto the space of coordinates y_{i_{1}}, …, y_{i_{d}}, y_{l+1}, …, y_{n} is of dimension less than the dimension n − l + d of the ambient space. Hence there is a non-zero polynomial

h_{i_{1},...,i_{d},l+1,...,n}(a, y_{i_{1}}, …, y_{i_{d}}, y_{l+1}, …, y_{n})

whose construction depends on i_{1}, …, i_{d}, l + 1, …, n but not on ζ, which is zero on V (a). Write the polynomial in the form

h_{i_{1},...,i_{d},l+1,...,n} = ∑_{m∈M_{i_{1},...,i_{d}}} g_{m,i_{1},...,i_{d},l+1,...,n}(a, y_{i_{1}}, …, y_{i_{d}}) · ⟨y_{l+1}, …, y_{n}⟩^{m},

where M_{i_{1},...,i_{d}} is a set of (n − l)-tuples ⟨m_{l+1}, …, m_{n}⟩ of integers, g_{m,i_{1},...,i_{d},l+1,...,n} non-zero polynomials, and ⟨y_{l+1}, …, y_{n}⟩^{m} denotes y_{l+1}^{m_{l+1}} ·. . .·y_{n}^{m_{n}}. Notice that M_{i_{1},...,i_{d}} contains at least two distinct elements for otherwise there is a component of projective completion V̄ (a) of V (a) in the subvariety y_{i} = 0 or y_{i}^{−1} = 0 for some i of P^{n}(C). The next step in the algorithm depends on the cases. Case 1: ⟨ζ_{l+1}, …, ζ_{n}⟩^{m−m′}/D is either 0 or ∞ for all distinct m, m′ ∈ M_{i_{1},...,i_{d}} Case 2: ⟨ζ_{l+1}, …, ζ_{n}⟩^{m−m′}/D is in C^{∗} for some distinct m, m′ ∈ M_{i_{1},...,i_{d}}. In the second case, assuming m_{l+1}−m′_{l+1} ≠ 0 we change the variables x_{i}(new) = x_{i} for all i ≠ l+1 and x_{l+1}(new) = (m_{l+1}−m′_{l+1})·x_{l+1}+…+(m_{n}−m′_{n})·x_{n}. Correspondingly, y_{i}(new) = exp(x_{i}(new)). Then in the new variables ζ_{1}/D, …, ζ_{l+1}/D are in C^{∗}, ζ_{l+2}/D = … = ζ_{n}/D = 0, and we come to the first step of the algorithm with l + 1 instead of l. In case 1 find m_{0,i_{1},...,i_{d},l+1,...,n} ∈ M_{i_{1},...,i_{d}} which satisfies the ”minimality” condition:

⟨ζ_{l+1}, …, ζ_{n}⟩^{m−m_{0,i_{1},...,i_{d},l+1,...,n}}/D = 0 for all m ∈ M_{i_{1},...,i_{d}} distinct from m_{0,i_{1},...,i_{d},l+1,...,n}.

Since

h_{i_{1},...,i_{d},l+1,...,n}(a, ζ/D) = 0

we get necessarily

g_{m_{0,i_{1},...,i_{d},l+1,...,n}}(a, ζ_{i_{1}}/D, …, ζ_{i_{d}}/D) = 0.

Notice that this case is not possible for d = 0 since the polynomial has not been identically zero. We terminate the algorithm and define W (a) to be the algebraic subvariety of S = C^{l} determined by the set of equations

{ ∏_{m_{0,i_{1},...,i_{d},l+1,...,n}∈M_{i_{1},...,i_{d}}} g_{m_{0,i_{1},...,i_{d},l+1,...,n}}(a, y_{i_{1}}, …, y_{i_{d}}) = 0 : i_{1} < … < i_{d} ≤ l}

in variables y_{1}, …, y_{l}. Notice that the length of the algorithm is restricted by n and on every step we choose between finitely many possibilities. Thus there are only finitely many outcomes of the algorithm.

Also, dim W (a) < d since any d coordinates on W (a) are algebraically dependent. Hence, by the choice of d, l − dim W (a) > n − dim V (a). Consider now the projection pr_{l}(L) = pr_{1,...l}(L) of L into C^{l} in coordinates x_{1}, …, x_{l} along x_{l+1}, …, x_{n}. Notice that the mapping

pr_{1} : L → pr_{l}(L)

has a non-trivial kernel. Otherwise the mapping is the linear isomorphism and hence for l < i ≤ n there are real s_{i,1}, …, s_{i,l} ∈ K such that for any ⟨x_{1}, …, x_{n}⟩ ∈ L

x_{i} = s_{i,1} · x_{1} + … + s_{i,l} · x_{l}.

It follows

ζ_{i} = exp(s_{i,1} · ln ζ_{1}) · . . . · exp(s_{i,l} · ln ζ_{l}),

and going to limit we obtain the contradiction with the fact that, for l < i ≤ n, j ≤ l, by the construction ζ_{i}/D = 0, while exp(s_{i,j} · ln ζ_{j}/D) are non-zero. Since the kernel is non-trivial, dim pr_{l}(L) < dim L.□

Remark The varieties W (a) constructed in Lemma above are just shortenings of V (a) in the terminology of [Kh].

Notation Denote for a subset X ⊆ C^{k}

Re(X) = {Re(x) : x ∈ X}.

Notice that Re(L) under the assumption that K ⊆ R is an R-linear subspace of R^{k}.

Lemma 1.3 There are a real number R = R(L, V) and a positive integer m = m(L, V) satisfying the following: Given a ∈ C(V), there is a set μ(L, V, a) of m R-affine hyperplanes of Re(L), such that for any ball B ⊆ Re(L) of radius R which intersect no hyperplane from μ(L, V, a) there is a point in (Re(L) + ıB) ∩ L ∩ ln V (a).

Proof By [Kh, section 6] taking into account Lemma 1.1, for any closed ball B ⊆ Re(L) of a large enough radius the set

C_{B}(L, V) = {a ∈ C(V) : L ∩ ln V (a) is non-degenerate in Re(L) + ıB}

is dense in C(V) (in the real topology) and any exponential system in the family C_{B}(L, V) has a solution in Re(L) + ıB. Suppose a ∈ C(L, V). Then a = a_{i}/D the limit point of a sequence {a_{i} ∈ C_{B}(V) : i ∈ N} along an ultrafilter D on N. By the above for each i one can find a solution

ξ(i) ∈ (Re(L) + ıB) ∩ ln V (a_{i})

(notice that Re(L) + ıB ⊆ L). Correspondingly, exp(ξ(i)) = ζ(i) ∈ L ∩ V (a_{i}). The limit ξ/D of the sequence ξ(i) is either a point in (Re(L) + ıB) ∩ ln V (a) or ζ/D = exp(ξ/D) satisfies the conditions of Lemma 1.2 (iii). Then pr_{l}(ζ/D) ∈ exp(pr_{l}(L)) ∩ W (a) for one of finitely many W ∈ ρ(L, V). Thus

pr_{l}(ξ/D) ∈ pr_{l}(Re(L) + ıB) ∩ ln W (a).

It follows from (ii) of Lemma 1.2 that any point in the intersection is atypical. Thus by Theorem 1 pr_{l}(ξ/D) must lie in one of finetely many affine hyperplanes of pr_{l}(L) of the form H + c_{i} for H ∈ η(pr_{l}(L), W) and i ≤ p(pr_{l}(L), W). Thus B must intersect one of the hyperplanes of Re(L) of the form Re[pr_{l}^{−1}(H + c_{i})]. Take all such hyperplanes to be μ(L, V, a). □

Lemma 1.4 Given a natural number m, there is a real constant R^{∗}(m) such that given a ball B^{∗} in Re(L) of radius R^{∗}(m) and a set H of m affine hyperplanes in the space there is a ball B ⊆ B^{∗} of radius R which does not intersect with any hyperplane of H.

Proof By induction on m one easily sees that R^{∗}(m) = 2^{m} · R fits.□

Theorem 2 Assume SchC+CIT. Let L ⊆ C^{n} be an R-linear subspace and V a family of algebraic varieties given by equations (1) such that (L, V (a)) is normal and free for any a ∈ C(V). Then there is a constant R(L,V) such that, given a ball B^{∗} of radius R(L, V), there is always a solution x of the exponential system with

x ∈ (Re(L) + ıB^{∗}).

Proof Take R = R(V, L) = R^{∗}(m(L, V)) of Lemma 1.4. Then for any B^{∗} of radius R there is B ⊆ B^{∗} such that B does not intersect any hyperplane from μ(L, V, a). Apply now Lemma 1.3.□

Remark The theorem can be strengthened to fulfill the requirement that, given any extra l R-affine hyperplanes, we can choose R(V, L, l) so that one can find a solution in Re(L) + ıB^{∗} outside the l hyperplanes.

References

[GW] P.M.Gruber, J.M.Wills (editors) Handbook of Convex Geometry,

[K] B.Kazarnovski, Exponential analytic sets, Functional Analysis and Applications, 1997, v.31, no.2, 86-94

[Kh] A.Khovanski, Fewnomials,(Russian edition), Fazis, Moscow, 1997

[Z1] B.Zilber, Intersecting varieties with tori, Prepublication, 2000

[Z2] B.Zilber, Raising to powers in algebraically closed fields, In preparation