Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Generalized product property

TL;DR

This paper generalizes Siciak's classical result relating extremal functions of sets to their Cartesian product, extending it to functions with growth controlled by convex sets, and explores limitations in the weighted case.

Contribution

The paper extends Siciak's product property to Siciak-Zakharyuta functions with growth in convex sets, broadening the understanding of extremal functions in pluripotential theory.

Findings

Generalization of Siciak's result to convex set growth functions

Identification of limitations in weighted extremal functions

Insights into pluripotential theory and extremal functions

Abstract

A famous result of Siciak is how the Siciak-Zakharyuta functions, sometimes called global extremal functions or pluricomplex Green functions with a pole at infinity, of two sets relate to the Siciak-Zakharyuta function of their cartesian product. In this paper Siciak's result is generalized to the setting of Siciak-Zakharyuta functions with growth given by a compact convex set, along with discussing why this generalization does not work in the weighted setting.