Gauss-Lucas theorems for entire functions on CM

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 sections can be written as a monomial times a canonical product of arbitrary genus. A connection is made with the Levy-Steinitz theorem for conditionally convergent vector series, a result generalizing Riemann’s well-known theorem for conditionally convergent real number series.