Polynomials with exponents in compact convex sets and associated weighted extremal functions -- The Siciak-Zakharyuta theorem

2023/05/14 by Magnússon, Benedikt Steinar, Sigurðardóttir, Álfheiður Edda, Sigurðsson, Ragnar
#32A15 #32U15 #32U35 (Primary) 32A08 #32W05 (Secondary) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2305.08260

Abstract

The classical Siciak-Zakharyuta theorem states that the Siciak-Zakharyuta function V_E of a subset E of \mathbb Cⁿ, also called a pluricomplex Green function or global exremal function of E, equals the logarithm of the Siciak function Φ_E if E is compact. The Siciak-Zakharyuta function is defined as the upper envelope of functions in the Lelong class that are negative on E, and the Siciak function is the upper envelope of m-th roots of polynomials p in P_m(\mathbb Cⁿ) of degree ≤ m such that |p|≤ 1 on E. We generalize the Siciak-Zakharyuta theorem to the case where the polynomial space \mathcal P_m(\mathbb Cⁿ) is replaced by \mathcal P_m^S(\mathbb Cⁿ) consisting of all polynomials with exponents restricted to sets mS, where S is a compact convex subset of \mathbb Rⁿ₊ with 0∈ S. It states that if q is an admissible weight on a closed set E in \mathbb Cⁿ then V^S_E,q=logΦ^S_E,q on \mathbb C^*n if and only if the rational points in S form a dense subset of S.

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