Polynomials with exponents in compact convex sets and associated weighted extremal functions – fundamental results
Volume 133 / 2024
Annales Polonici Mathematici 133 (2024), 37-70
MSC: Primary 32U35; Secondary 32A08, 32A15, 32U15, 32W05
DOI: 10.4064/ap240223-4-10
Published online: 4 November 2024
Abstract
This paper is a collection of fundamental results about the polynomial rings where the mth degree polynomials have exponents restricted to mS, where S\subseteq \mathbb R^n_+ is compact, convex and 0\in S. We study the relationship between \mathcal P^S(\mathbb C^n) and the class \mathcal L^S(\mathbb C^n) of global plurisubharmonic functions where the growth is determined by the logarithmic supporting function of S. We present the properties of their respective weighted extremal functions \Phi _{K, q}^S and V_{K, q}^S in connection with the properties of S. Our ambition is to give detailed proofs with minimal assumptions of all results, thus creating a self-contained exposition.
