TY - JOUR

T1 - On Zeroes of Random Polynomials and an Application to Unwinding

AU - Steinerberger, Stefan

AU - Wu, Hau Tieng

N1 - Publisher Copyright:
© 2019 The Author(s). Published by Oxford University Press. All rights reserved.

PY - 2021/7/1

Y1 - 2021/7/1

N2 - 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.

AB - 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.

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U2 - 10.1093/imrn/rnz096

DO - 10.1093/imrn/rnz096

M3 - Article

AN - SCOPUS:85122290339

SN - 1073-7928

VL - 2021

SP - 10100

EP - 10117

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 13

ER -