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A New Lower Bound for the Maximal Valence of Harmonic Polynomials

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Abstract

We find a new lower bound for the maximal number of zeros of harmonic polynomials, \(p(z)+\overline{q(z)}\), when \(\deg p = n\) and \(\deg q = n-2\).

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Notes

  1. For \((n,m)=(4,2)\) the maximal valence achieved so far is 12. The maximal valence for even n needs to be even due to the argument principle. See, for example, [1].

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Acknowledgments

The authors would like to thank Prof. Dmitry Khavinson and Prof. Catherine Bénéteau for the motivating discussions in the early stage of the project. The first author was supported by Simons Collaboration Grants for Mathematicians.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of South Florida, 4202 East Fowler Ave, CMC 342, Tampa, FL, 33620, USA

    Seung-Yeop Lee & Andres Saez

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  1. Seung-Yeop Lee
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  2. Andres Saez
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Correspondence to Seung-Yeop Lee.

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Communicated by Stephan Ruscheweyh.

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Lee, SY., Saez, A. A New Lower Bound for the Maximal Valence of Harmonic Polynomials. Comput. Methods Funct. Theory 17, 139–149 (2017). https://doi.org/10.1007/s40315-016-0175-x

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  • DOI: https://doi.org/10.1007/s40315-016-0175-x

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