PAIRING BETWEEN ZEROS AND CRITICAL POINTS OF RANDOM POLYNOMIALS WITH INDEPENDENT ROOTS

摘要

Let p(n) be a random, degree n polynomial whose roots are chosen independently according to the probability measure mu on the complex plane. For a deterministic point lying outside the support of mu, we show that almost surely the polynomial q(n)(z) := p(n)(z)(z - xi) has a critical point at distance O(1/n) from xi. In other words, conditioning the random polynomials p(n) to have a root at xi almost surely forces a critical point near xi. More generally, we prove an analogous result for the critical points of q(n)(z) := p(n)(z)(z - xi(1)) ... (z - xi(k)), where xi(1), ... , xi(k) are deterministic. In addition, when k = o(n), we show that the empirical distribution constructed from the critical points of q(n) converges to mu in probability as the degree tends to infinity, extending a recent result of Kabluchko [Proc. Amer. Math. Soc. 143 (2015), no. 2, 695-702].