On Zeroes of Random Polynomials and an Application to Unwinding

Stefan Steinerberger, Hau Tieng Wu

Research output: Contribution to journalArticlepeer-review

Abstract

Let μ be a probability measure in C with a continuous and compactly supported density function, let z1, zn be independent random variables, zi ∼ μ, and consider the random polynomial pn(z) = ∏ k=1n(z-zk).We determine the asymptotic distribution of left z C: pn(z) = pn(0). In particular, if mu is radial around the origin, then those solutions are also distributed according to mu as n. Generally, the distribution of the solutions will reproduce parts of mu and condense another part on curves. We use these insights to study the behavior of the Blaschke unwinding series on random data.

Original languageEnglish (US)
Pages (from-to)10100-10117
Number of pages18
JournalInternational Mathematics Research Notices
Volume2021
Issue number13
DOIs
StatePublished - Jul 1 2021

ASJC Scopus subject areas

  • General Mathematics

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