Equidistribution of Zeros of Random Polynomials and Random Polynomial mappings on $\mathbb{C}^m$

TL;DR

This paper proves a general equidistribution of zeros for random polynomials in complex spaces, extending previous results to broader probabilistic settings without requiring i.i.d. coefficients.

Contribution

It establishes a new, more general equidistribution theorem for zeros of random polynomials, applicable even without i.i.d. coefficients and for higher codimension cases.

Findings

Equidistribution holds without i.i.d. coefficients.

Results apply to various probability distributions including Gaussian.

Extension to higher codimension polynomial bases.

Abstract

We study 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 of Bayraktar, Bloom and Levenberg and have 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 result turns out to be true without the random coefficients being i.i.d. (independent and identically distributed), which also means that there is no need to use any specific probability distribution function for these random coefficients. In the last section, unlike from 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 including the (standard) Gaussian and the Fubini-Study probability distributions as special cases, we have an equidistribution result for codimensions bigger than $1$.