  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.
