Real geometric transcendence for the Gamma function

TL;DR

This paper proves that the only real algebraic curve mapped into an algebraic curve by the Gamma function is the x-axis, using complex transcendence results and base-change techniques.

Contribution

It establishes a new real geometric transcendence result for the Gamma function, linking it to complex transcendence and applications to Manin--Mumford type conjectures.

Findings

The x-axis is the only real algebraic curve mapped into an algebraic curve by Gamma.

Employs Tamiozzo's base-change argument to derive real results from complex transcendence.

Provides applications to analogues of the Manin--Mumford conjecture for Gamma.

Abstract

We show that the $x$-axis is the only real algebraic curve in $\mathbb R^2$ whose image via the Gamma function is contained in an algebraic curve. Our proof employs an elegant base-change argument due to Tamiozzo (2023) to deduce the result from the corresponding complex geometric transcendence result of Eterovi\'c, Padgett and Zhao (2025). As an application, we use the complex and real geometric transcendence results to study analogues of the Manin--Mumford conjecture for the Gamma function.