Bialgebraic varieties of the Gamma function

Abstract

We characterize the bialgebraic varieties of the $\Gamma$ function, that is, if $V,W\subseteq {\mathbb{C}}^n$ are irreducible affine algebraic variety which satisfy $\dim V=\dim W$ and $\Gamma (V)\subseteq W$, then the equations defining $V$ (and hence also $W$) either give an equality between coordinates, or set some coordinates to be constant. We also classify the case where $V$ and $W$ have the same dimension as varieties over the field of algebraic functions.

17 pages. New version includes a second main theorem