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On Zeroes of Random Polynomials and an Application to Unwinding

Publication ,  Journal Article
Steinerberger, S; Wu, HT
Published in: International Mathematics Research Notices
July 1, 2021

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.

Duke Scholars

Published In

International Mathematics Research Notices

DOI

EISSN

1687-0247

ISSN

1073-7928

Publication Date

July 1, 2021

Volume

2021

Issue

13

Start / End Page

10100 / 10117

Related Subject Headings

  • General Mathematics
  • 4902 Mathematical physics
  • 0101 Pure Mathematics
 

Citation

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ICMJE
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Steinerberger, S., & Wu, H. T. (2021). On Zeroes of Random Polynomials and an Application to Unwinding. International Mathematics Research Notices, 2021(13), 10100–10117. https://doi.org/10.1093/imrn/rnz096
Steinerberger, S., and H. T. Wu. “On Zeroes of Random Polynomials and an Application to Unwinding.” International Mathematics Research Notices 2021, no. 13 (July 1, 2021): 10100–117. https://doi.org/10.1093/imrn/rnz096.
Steinerberger S, Wu HT. On Zeroes of Random Polynomials and an Application to Unwinding. International Mathematics Research Notices. 2021 Jul 1;2021(13):10100–17.
Steinerberger, S., and H. T. Wu. “On Zeroes of Random Polynomials and an Application to Unwinding.” International Mathematics Research Notices, vol. 2021, no. 13, July 2021, pp. 10100–17. Scopus, doi:10.1093/imrn/rnz096.
Steinerberger S, Wu HT. On Zeroes of Random Polynomials and an Application to Unwinding. International Mathematics Research Notices. 2021 Jul 1;2021(13):10100–10117.
Journal cover image

Published In

International Mathematics Research Notices

DOI

EISSN

1687-0247

ISSN

1073-7928

Publication Date

July 1, 2021

Volume

2021

Issue

13

Start / End Page

10100 / 10117

Related Subject Headings

  • General Mathematics
  • 4902 Mathematical physics
  • 0101 Pure Mathematics