Some remarks on Schanuel's conjecture

Abstract

Schanuel's Conjecture is the statement: if x 1,…,x n ∈ C are linearly independent over Q , then the transcendence degree of Q(x 1,…,x n, exp(x 1),…, exp(x n)) over Q is at least n. Here we prove that this is true if instead we take infinitesimal elements from any ultrapower of C , and in fact from any nonarchimedean model of the theory of the expansion of the field of real numbers by restricted analytic functions.