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Level Sets, a Gauss-Fourier Conjecture, and a Counter-Example to a Conjecture of Borcea and Shapiro

Computational Methods and Function Theory volume 11, pages 1–12 (2011)Cite this article

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 Im Q(z) > 0 where Q = z− f/f ′ and f is a real entire function.

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Authors and Affiliations

  1. Department of Mathematics, Hope College, Holland, MI, 49423, USA

    Stephanie Edwards

  2. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Aimo Hinkkanen

Authors
  1. Stephanie Edwards
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  2. Aimo Hinkkanen
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Corresponding author

Correspondence to Stephanie Edwards.

Additional information

This material is based upon work supported by the National Science Foundation under Grant No. 0758226. This research was also supported by the grant 07365 from the Campus Research Board of the University of Illinois at Urbana—Champaign.

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Edwards, S., Hinkkanen, A. Level Sets, a Gauss-Fourier Conjecture, and a Counter-Example to a Conjecture of Borcea and Shapiro. Comput. Methods Funct. Theory 11, 1–12 (2011). https://doi.org/10.1007/BF03321786

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  • DOI: https://doi.org/10.1007/BF03321786

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