Level sets, a Gauss-Fourier conjecture, and a counter-example to a conjecture of Borcea and Shapiro

Stephanie Edwards, Aimo Hinkkanen

Research output: Contribution to journalArticlepeer-review

Abstract

We use level sets to give a counter-example to a conjecture of J. Borcea and B. Shapiro. In particular we show that there are real polynomials P such that there is a chord of the level set Im P'(z)=P(z) = 0 that does not have a non-real zero of P lying on it. In addition, we use level sets to show that there are no bounded components of the set {z: Im z > 0 and ImQ(z) > 0} where Q = z -f/f" and f is a real entire function.

Original languageEnglish (US)
Pages (from-to)1-12
Number of pages12
JournalComputational Methods and Function Theory
Volume11
Issue number1
DOIs
StatePublished - Jan 1 2011

Keywords

  • Hawai'i Conjecture
  • Level sets
  • Real polynomials
  • Zeros

ASJC Scopus subject areas

  • Analysis
  • Computational Theory and Mathematics
  • Applied Mathematics

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