Equidistribution of zeros of random polynomials and random polynomial mappings on $\pmb{\mathbb{C}}^{m}$

O. Günyüz

Sbornik. Mathematics, Tome 216 (2025) no. 1, pp. 54-70

Résumé

We study the equidistribution problem of zeros in relation to a sequence of $Z$-asymptotically Chebyshev polynomials on $\mathbb{C}^{m}$. We use certain results obtained in a very recent work by Bayraktar, Bloom and Levenberg and obtain an equidistribution result in a more general probabilistic setting than what the paper of Bayraktar, Bloom and Levenberg considers, even though the basis polynomials they use are more general than $Z$-asymptotically Chebyshev polynomials. Our equidistribution result is based on the expected distribution and the variance estimate of random zero currents corresponding to the zero sets (zero divisors) of polynomials. This equidistribution result of general nature shows that equidistribution turns out to be true without the random coefficients being independent and identically distributed, which also means that there is no need to use any specific probability distribution function for these random coefficients. In § 3, unlike in the $1$-codimensional case, we study the basis of polynomials orthogonal with respect to the $L^{2}$-inner product defined by the weighted asymptotically Bernstein–Markov measures on a given locally regular compact set, and with a probability distribution studied well by Bayraktar and including the (standard) Gaussian and the Fubini–Study probability distributions as special cases we have an equidistribution result for codimensions larger than $1$. Bibliography: 35 titles.

Détail

Keywords: random polynomials, equidistribution of zeros, variance, Chebyshev polynomials.