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

Let pn be a random, degree n polynomial whose roots are chosen independently according to the probability measure μ on the complex plane. For a deterministic point ξ lying outside the support of μ, we show that almost surely the polynomial qn(z) := pn(z)(z − ξ) has a critical point at distance O(1/n) from ξ. In other words, conditioning the random polynomials pn to have a root at ξ almost surely forces a critical point near ξ. More generally, we prove an analogous result for the critical points of qn(z) := pn(z)(z − ξ1) · · · (z−ξk), where ξ1, . . ., ξk are deterministic. In addition, when k = o(n), we show that the empirical distribution constructed from the critical points of qn converges to μ in probability as the degree tends to infinity, extending a recent result of Kabluchko. [ABSTRACT FROM AUTHOR]