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.
Categories: math.NT