Pairing of zeros and critical points for random polynomials

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Abstract

Let pN be a random degree N polynomial in one complex variable whose zeros are chosen independently from a fixed probability measure μ on the Riemann sphere S2. This article proves that if we condition pN to have a zero at some fixed point ζ ϵ S2, then, with high probability, there will be a critical point ωζ at a distance N-1 away from ζ. This N-1 distance is much smaller than the N-1/2 typical spacing between nearest neighbors for N i.i.d. points on S2. Moreover, with the same high probability, the argument of ωζ relative to ζ is a deterministic function of μ plus fluctuations on the order of N-1.

Original languageEnglish (US)
Pages (from-to)1498-1511
Number of pages14
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume53
Issue number3
DOIs
StatePublished - Aug 2017
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Critical points
  • Gauss-Lucas
  • Random polynomials
  • Zeros

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