### Zeros of random polynomials

- Series
- School of Mathematics Colloquium
- Time
- Thursday, October 23, 2014 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Professor Igor Pritsker – Oklahoma State University

The area was essentially originated by the general question: How many
zeros of a random polynomials are real? Kac showed that the expected
number of real zeros for a polynomial with i.i.d. Gaussian coefficients
is logarithmic in terms of the degree. Later, it was found that most of
zeros of random polynomials are asymptotically uniformly distributed
near the unit circumference (with probability one) under mild
assumptions on the coefficients.
Thus two main directions of research are related to the almost sure
limits of the zero counting measures, and to the quantitative results on
the expected number of zeros in various sets. We give estimates of the
expected discrepancy between the zero counting measure and the
normalized arclength on the unit circle. Similar results are established
for polynomials with random coefficients spanned by various bases,
e.g., by orthogonal polynomials. We show almost sure convergence of the
zero counting measures to the corresponding equilibrium measures for
associated sets in the plane, and quantify this convergence. Random
coefficients may be dependent and need not have identical distributions
in our results.