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Exponential sums equations and the Schanuel conjecture. (English) Zbl 1030.11073
Following certain ideas and constructions in model theory, we discuss here a ‘uniform’ version of the Schanuel conjecture: for any algebraic variety \(V\subseteq \mathbb C^{2n}\) of dimension less than \(n\) there is a finite set \(\mu(V)\) of proper \(\mathbb Q\)-linear subspaces of \(\mathbb C^n\) such that, given \[ \langle x_1,\dots , x_n,\exp(x_1),\dots ,\exp(x_n)\rangle \in V, \] there is \(M\in \mu(V)\) and an integer vector \(\overline z\in \mathbb Z^n\) such that \[ \langle x_1,\dots , x_n\rangle\in M+ 2\pi i\cdot \overline z, \] and \(M\) is of codimension at least 2 or \(\overline z=0.\)
As a matter of fact, this conjecture is a combination of the Schanuel conjecture and a conjecture of a Diophantine kind about intersections of algebraic varieties and tori. The latter can be naturally generalized to the case of semi-Abelian varieties, and in this generality it is stronger than the statement of the Mordell-Lang conjecture.
Assuming the uniform Schanuel conjecture we prove that any ‘non-obviously-contradictory’ system of equations in the form of exponential sums with real exponents [see B. Ya. Kazarnovskij, Funct. Anal. Appl. 31, No. 2, 86-94 (1997; Zbl 0911.32008)] has a solution.

11U09 Model theory (number-theoretic aspects)
11J99 Diophantine approximation, transcendental number theory
03C65 Models of other mathematical theories
11L99 Exponential sums and character sums
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