Zeros of random polynomials and its higher derivatives

Abstract

In this article we study the limiting empirical measure of zeros of higher derivatives for sequences of random polynomials. We show that these measures agree with the limiting empirical measure of zeros of corresponding random polynomials. Various models of random polynomials are considered by introducing randomness through multiplying a factor with a random zero or removing a zero at random for a given sequence of deterministic polynomials. We also obtain similar results for random polynomials whose zeros are given by i.i.d. random variables. As an application, we show that these phenomenon appear for random polynomials whose zeros are given by the 2D Coulomb gas density.

arXiv comment21 pages

References in corpus (6)

• Coulomb gas ensembles and Laplacian growth
• Asymptotic distribution of complex zeros of random analytic functions
• Concentration for Coulomb gases and Coulomb transport inequalities
• Roots of the derivative of the Riemann zeta function and of characteristic polynomials
• Pairing between zeros and critical points of random polynomials with independent roots
• The Common Limit of the Linear Statistics of Zeros of Random Polynomials and Their Derivatives

Cited by in corpus (2)

• On the local pairing behavior of critical points and roots of random polynomials
• Distances between zeroes and critical points for random polynomials with i.i.d. zeroes