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

O’Rourke, Sean; Williams, Noah

doi:10.1090/tran/7496

Pairing between zeros and critical points of random polynomials with independent roots
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by Sean O’Rourke and Noah Williams PDF

Trans. Amer. Math. Soc. 371 (2019), 2343-2381 Request permission

Abstract:

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 $O(1/n)$ 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 $\xi _1, \ldots , \xi _k$ 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].

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