Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Fundamental results
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| Format: | Preprint |
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2023
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| author | Magnússon, Benedikt Steinar Sigurðardóttir, Álfheiður Edda Sigurðsson, Ragnar Snorrason, Bergur |
| author_facet | Magnússon, Benedikt Steinar Sigurðardóttir, Álfheiður Edda Sigurðsson, Ragnar Snorrason, Bergur |
| contents | This paper is a survey of plurisubharmonic theory where the usual polynomial ring is replaced by a polynomial ring $\mathcal P^S(\mathbb C^n)$ where the $m$-th degree polynomials have exponents restricted to $mS$, where $S\subseteq \mathbb R^n_+$ is compact, convex and $0\in S$. We assume no other conditions on $S$ as these are necessary for $\mathcal P^S(\mathbb C^n)$ to be a graded polynomial ring. 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 properties of their respective weighted extremal functions $Φ_{K, q}^S$ and $V_{K, q}^S$ in connection with properties of $S$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_04779 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Fundamental results Magnússon, Benedikt Steinar Sigurðardóttir, Álfheiður Edda Sigurðsson, Ragnar Snorrason, Bergur Complex Variables 32U35 (Primary) 32A08, 32A15, 32U15, 32W05 (Secondary) This paper is a survey of plurisubharmonic theory where the usual polynomial ring is replaced by a polynomial ring $\mathcal P^S(\mathbb C^n)$ where the $m$-th degree polynomials have exponents restricted to $mS$, where $S\subseteq \mathbb R^n_+$ is compact, convex and $0\in S$. We assume no other conditions on $S$ as these are necessary for $\mathcal P^S(\mathbb C^n)$ to be a graded polynomial ring. 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 properties of their respective weighted extremal functions $Φ_{K, q}^S$ and $V_{K, q}^S$ in connection with properties of $S$. |
| title | Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Fundamental results |
| topic | Complex Variables 32U35 (Primary) 32A08, 32A15, 32U15, 32W05 (Secondary) |
| url | https://arxiv.org/abs/2305.04779 |