On the distribution of polynomials with bounded roots, I. Polynomials with real coefficients

Shigeki Akiyama, Attila Petho

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

Let v(s) d denote the set of coefficient vectors of contractive polynomials of degree d with 2s non-real zeros. We prove that v(s) d can be computed by a multiple integral, which is related to the Selberg integral and its generalizations. We show that the boundary of the above set is the union of finitely many algebraic surfaces. We investigate arithmetical properties of v(s) d and prove among others that they are rational numbers. We will show that within contractive polynomials, the 'probability' of picking a totally real polynomial decreases rapidly when its degree becomes large.

Original languageEnglish
Pages (from-to)927-949
Number of pages23
JournalJournal of the Mathematical Society of Japan
Volume66
Issue number3
DOIs
Publication statusPublished - 2014

Keywords

  • Polynomials with bounded roots
  • Selberg integral

ASJC Scopus subject areas

  • Mathematics(all)

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