Pairing between zeros and critical points of random polynomials with independent roots

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 $\xi$ 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 $O(1\mathrm{/}n)$ 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 $q_n(z):=p_n(z)(z-{\xi }_1)\cdots (z-{\xi }_k)$, where $\xi_1, \ldots, \xi_k$aredeterministic\mathrm{.}Inaddition,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.