Zero-free neighborhoods around the unit circle for Kac polynomials

Gerardo Barrera, Paulo Manrique-Mirón

Research output: Contribution to journalArticleScientificpeer-review


In this paper, we study how the roots of the Kac polynomials concentrate around the unit circle when the coefficients are independent and identically distributed non-degenerate real random variables. It is well-known that the roots of a Kac polynomial concentrate around the unit circle as the dimension growths if and only if some log-moment is finite. Under the finiteness of the second moment, we show that there exists an annulus of small width around the unit circle which is free of roots with high probability. The proof relies on small ball probability inequalities and the least common denominator.

Original languageEnglish
JournalPeriodica Mathematica Hungarica
Issue number2
Pages (from-to)159–176
Number of pages18
Publication statusPublished - Jun 2022
MoE publication typeA1 Journal article-refereed

Fields of Science

  • 111 Mathematics
  • Salem–Zygmund type inequalities
  • Zeros of random polynomials
  • 112 Statistics and probability
  • Small ball probability
  • Locally sub-Gaussian random variables
  • Locally sub-Gaussian random variables
  • Salem-Zygmund type inequalities
  • Small ball probability
  • Zeros of random polynomials
  • Complex zeros

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