Bayraktar, Turgay and Bloom, Thomas and Levenberg, Norman
(2023)
*Random polynomials in several complex variables.*
Journal d'Analyse Mathematique
.
ISSN 0021-7670 (Print) 1565-8538 (Online)
Published Online First http://dx.doi.org/10.1007/s11854-023-0316-x

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Official URL: http://dx.doi.org/10.1007/s11854-023-0316-x

## Abstract

We generalize some previous results on random polynomials in several complex
variables. A standard setting is to consider random polynomials $H_n(z):=\sum_{j=1}^{m_n} a_jp_j(z)$ that are linear combinations of basis polynomials $\{p_j\}$ with i.i.d. complex random variable coefficients $\{a_j\}$ where $\{p_j\}$ form an orthonormal basis for a Bernstein-Markov measure on a compact set $K\subset \C^d$. Here $m_n$ is the dimension of $\mathcal P_n$, the holomorphic polynomials of degree at most $n$ in $\C^d$. We consider more general bases $\{p_j\}$, which include, e.g., higher-dimensional generalizations of Fekete polynomials. Moreover we allow $H_n(z):=\sum_{j=1}^{m_n} a_{nj}p_{nj}(z)$; i.e., we have an {\it array} of basis polynomials $\{p_{nj}\}$ and random coefficients $\{a_{nj}\}$. This always occurs in a weighted situation. We prove results on convergence in probability and on almost sure convergence of $\frac{1}{n}\log |H_n|$ in $L^1_{loc}(\C^d)$ to the (weighted) extremal plurisubharmonic function for $K$. We aim for weakest possible sufficient conditions on the random coefficients to guarantee convergence.

Item Type: | Article |
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Subjects: | Q Science > QA Mathematics Q Science > QA Mathematics > QA299.6-433 Analysis |

Divisions: | Faculty of Engineering and Natural Sciences > Basic Sciences > Mathematics Faculty of Engineering and Natural Sciences |

Depositing User: | Turgay Bayraktar |

Date Deposited: | 08 Feb 2024 15:19 |

Last Modified: | 08 Feb 2024 15:20 |

URI: | https://research.sabanciuniv.edu/id/eprint/48857 |