Gauss-Lucas Theorems for Entire Functions on CM

TL;DR

This paper extends the Gauss-Lucas theorem to multivariate entire functions using separate convexity, providing sharp results beyond the univariate case, and employs the Levy-Steinitz theorem for series convergence.

Contribution

It introduces a multivariate Gauss-Lucas theorem for entire functions with specific structural properties, expanding the scope beyond previous univariate restrictions.

Findings

Proves a multivariate Gauss-Lucas theorem using separate convexity.

Applies to entire functions with one-dimensional sections as monomials times canonical products.

Utilizes Levy-Steinitz theorem for series convergence in the proof.

Abstract

A Gauss-Lucas theorem is proved for multivariate entire functions, using a natural notion of separate convexity to obtain sharp results. Previous work in this area is mostly restricted to univariate entire functions (of genus no greater than one unless "realness" assumptions are made.) The present work applies to multivariate entire functions whose one dimensional sections can be written as a monomial times a canonical product of arbitrary genus. Essential use is made of the Levy-Steinitz theorem for conditionally convergent real number series.