Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Fundamental results

2023/05/08 by Magnússon, Benedikt Steinar, Sigurðardóttir, Álfheiður Edda, Sigurðsson, Ragnar +1
#32A15 #32U15 #32U35 (Primary) 32A08 #32W05 (Secondary) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2305.04779

Abstract

This paper is a survey of plurisubharmonic theory where the usual polynomial ring is replaced by a polynomial ring \mathcal P^S(\mathbb Cⁿ) where the m-th degree polynomials have exponents restricted to mS, where S⊆ \mathbb Rⁿ₊ is compact, convex and 0∈ S. We assume no other conditions on S as these are necessary for \mathcal P^S(\mathbb Cⁿ) to be a graded polynomial ring. We study the relationship between \mathcal P^S(\mathbb Cⁿ) and the class \mathcal L^S(\mathbb Cⁿ) of global plurisubharmonic functions where the growth is determined by the logarithmic supporting function of S. We present properties of their respective weighted extremal functions Φ_{K, q}^S and V_{K, q}^S in connection with properties of S.

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