Pairing of Zeros and Critical Points for Random Polynomials

TL;DR

This paper demonstrates that conditioning a random polynomial to have a zero at a fixed point results in a nearby critical point at a distance of 1/N, revealing a precise local pairing behavior between zeros and critical points.

Contribution

It establishes a probabilistic pairing between zeros and critical points of random polynomials conditioned on a zero, with detailed asymptotic behavior.

Findings

Critical points are typically within 1/N of conditioned zeros.

The argument of the critical point is a deterministic function plus small fluctuations.

The 1/N proximity is much smaller than typical zero spacing on the sphere.

Abstract

Let p_N be a random degree N polynomial in one complex variable whose zeros are chosen independently from a fixed probability measure mu on the Riemann sphere S^2. This article proves that if we condition p_N to have a zero at some fixed point xi in , then, with high probability, there will be a critical point w_xi a distance 1/N away from xi. This 1/N distance is much smaller than the one over root N typical spacing between nearest neighbors for N i.i.d. points on S^2. Moreover, with the same high probability, the argument of w_xi relative to xi is a deterministic function of mu plus fluctuations on the order of 1/N.