Schanuel Property for Elliptic and Quasi--Elliptic Functions

Abstract

For almost all tuples $(x_1,\dots ,x_n)$ of complex numbers, a strong version of Schanuel's Conjecture is true: the $2n$ numbers $x_1,\dots ,x_n,{\mathrm{e}}^{x_1},\dots ,{\mathrm{e}}^{x_n}$ are algebraically independent. Similar statements hold when one replaces the exponential function ${\mathrm{e}}^z$ with algebraically independent functions. We give examples involving elliptic and quasi--elliptic functions, that we prove to be algebraically independent: $z$, $\wp (z)$, $\zeta (z)$, $\sigma (z)$, exponential functions, and Serre functions related with integrals of the third kind.

26 pages