Zeros of random polynomials

Zeros of random polynomials

This workshop, sponsored by AIM and the NSF, will be devoted to the zero distribution of random polynomials spanned by various deterministic bases. The prototypical classical example is the Kac polynomials, where the coefficients are i.i.d. real (or complex) Gaussian random variables, and the basis is given by standard monomials. Recent trends include studies of more general ensembles of random polynomials with non-Gaussian coefficients that are spanned by various polynomial bases, e.g., trigonometric and orthogonal polynomials.

The main topics for the workshop are

• Global asymptotic distribution of zeros: We plan to study the limiting measures for the zero counting (empirical) measures, and provide the necessary and sufficient conditions on the coefficients and the basis functions to guarantee almost sure convergence. We also consider quantitative approaches to convergence of the zero counting measures either through the large deviation methods or the discrepancy results.
• Intensity functions for real and complex zeros: It is of interest to develop results on sharp asymptotics and explicit intensity functions for the zeros of random polynomials spanned by various bases with general random coefficients. It is known that different choices of bases and coefficients produce dramatically different phenomena in terms of intensity functions and the asymptotic behavior for the number of real and complex zeros.
• Local asymptotic results on zeros: We aim at the local universality (scaling) limits for the correlation functions of zeros of random polynomials here. Several recent results (such as the replacement principle of Tao and Vu) for the classical ensembles of random algebraic and trigonometric polynomials suggest methods for addressing local universality questions for general random orthogonal polynomials.

This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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