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Two Walsh-Type Theorems for the Solutions of Multi-Affine Symmetric Polynomials

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Progress in Approximation Theory and Applicable Complex Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 117))

Abstract

The spirit of the classical Grace-Walsh-Szegő coincidence theorem states that if there is a solution of a multi-affine symmetric polynomial in a domain with certain properties, then in it there exists another solution with other properties. We present two results in the same spirit, which may be viewed as extensions of the Grace-Walsh-Szegő result.

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References

  1. Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials. Oxford University Press Inc., New York (2002)

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  2. Sendov, Bl., Sendov, H.S.: Loci of complex polynomials, part I. Trans. Am. Math. Soc. 10 (366), 5155–5184 (2014)

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  3. Sendov, Bl., Sendov, H.S.: Loci of complex polynomials, part II: polar derivatives. Math. Proc. Camb. Phil. Soc. 159, 253–273 (2015)

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Acknowledgements

Our gratitude goes to an anonymous referee for his or her time and valuable comments that saved us from grave errors. The work of the first author has been partly supported by the Bulgarian National Science Fund grand i01/0006 and by EC FP7 Project AComIn - EP7-REGPOT-2012-2013-1. Hristo Sendov was partially supported by the Natural Sciences and Engineering Research Council of Canada.

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Authors and Affiliations

  1. Bulgarian Academy of Sciences, Institute of Information and Communication Technologies, Acad. G. Bonchev Str., bl. 25A, Sofia, 1113, Bulgaria

    Blagovest Sendov

  2. Department of Statistical and Actuarial Sciences, Western University, 1151 Richmond Str., London, ON, Canada, N6A 5B7

    Hristo Sendov

Authors
  1. Blagovest Sendov
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  2. Hristo Sendov
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Corresponding author

Correspondence to Hristo Sendov .

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Editors and Affiliations

  1. Department of Mathematics and Statistics, Auburn University, Auburn, Alabama, USA

    Narendra Kumar Govil

  2. Department of Mathematics, University of Central Florida, Orlando, Florida, USA

    Ram Mohapatra

  3. Department of Mathematics, Tuskegee University, Tuskegee, Alabama, USA

    Mohammed A. Qazi

  4. Department of Mathematics, University of Erlangen-Nuremberg, Erlangen, Germany

    Gerhard Schmeisser

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Sendov, B., Sendov, H. (2017). Two Walsh-Type Theorems for the Solutions of Multi-Affine Symmetric Polynomials. In: Govil, N., Mohapatra, R., Qazi, M., Schmeisser, G. (eds) Progress in Approximation Theory and Applicable Complex Analysis. Springer Optimization and Its Applications, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-49242-1_8

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