Comparing and Zilber’s Exponential Fields: Zero Sets of Exponential Polynomials

Abstract

We continue the research programme of comparing the complex exponential with Zilber's exponential. For the latter we prove, using diophantine geometry, various properties about zero sets of exponential functions, proved for C using analytic function theory, e.g. the Identity Theorem.

Generic solutions of equations with iterated exponentials

“…The analogous statement for the ring of power series tC [[t]] has been proved by Ax in [1]. Schanuel's Conjecture has played a crucial role in exponential algebra (see [15], [21], [3]), and in the model theory of exponential fields (see [16], [22], [18], [4], [5]). Assuming Schanuel's Conjecture, we are able to prove some particular cases of the Conjecture.…”

“…In this paper we consider the next natural cases of exponential polynomials with two and three iterations of exponentations, and we obtain an analogous result to that of Marker. Comparing the complex exponential field and Zilber's fields has been one of the main motivation in the following recent papers [3], [5], [8], [13].…”

Model theory of special subvarieties and Schanuel-type conjectures

“…P. D'Aquino, A.Macintyre and G.Terzo embarked on a programme of comparing F exp with C exp , in particular checking the validity of (EC) which they renamed the Nullstellensatz for exponential equation. The paper [14] proves that F exp contains a solution to any equation f (z) = 0, where f (z) is a onevariable term in exp, + and × with parameters, provided f (z) is not of the form exp(g(z)) for a term g(z). This was proved to hold in C exp by W. Henson and L. Rubin using Nevanlinna Theory, [18].…”

A weak version of the strong exponential closure

“…This would, of course, imply that C satisfies Schanuel's Conjecture (SC), and Zilber's Nullstellensatz. Comparing the complex exponential field and Zilber's fields has been object of study in [25], [7], [8], [24], [9].…”