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Borcea’s Variance Conjectures on the Critical Points of Polynomials

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Notions of Positivity and the Geometry of Polynomials

Part of the book series: Trends in Mathematics ((TM))

Abstract

Closely following recent ideas of J. Borcea, we discuss various modifications and relaxations of Sendov’s conjecture about the location of critical points of a polynomial with complex coefficients. The resulting open problems are formulated in terms of matrix theory, mathematical statistics or potential theory. Quite a few links between classical works in the geometry of polynomials and recent advances in the location of spectra of small rank perturbations of structured matrices are established. A couple of simple examples provide natural and sometimes sharp bounds for the proposed conjectures.

Mathematics Subject Classification (2000). Primary 12D10; Secondary 26C10, 30C10, 15A42, 15B05.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of South Florida, 4202 East Fowler Ave, PHY114, Tampa, FL, 33620-5700, USA

    Dmitry Khavinson

  2. Department of Mathematics and Statistics, University of Guelph, 50 Stone Road East, Guelph, Ontario, N1G 2W1, Canada

    Rajesh Pereira

  3. Department of Mathematics, University of California, Santa Barbara, CA, 93106, USA

    Mihai Putinar

  4. Center for Constructive Approximation Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA

    Edward B. Saff

  5. Department of Mathematics, Royal Institute of Technology, S-100 44, Stockholm, Sweden

    Serguei Shimorin

Authors
  1. Dmitry Khavinson
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  2. Rajesh Pereira
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  3. Mihai Putinar
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  4. Edward B. Saff
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  5. Serguei Shimorin
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Correspondence to Dmitry Khavinson .

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Editors and Affiliations

  1. , Department of Mathematics, Royal Institute of Technology, Stockholm, 100 44, Sweden

    Petter Brändén

  2. , Mathematical Department, Stockholm University, Stockholm, SE-106 91, Sweden

    Mikael Passare

  3. , Department of Mathematics, University of California, Santa Barbara, 93106-3080, California, USA

    Mihai Putinar

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Khavinson, D., Pereira, R., Putinar, M., Saff, E.B., Shimorin, S. (2011). Borcea’s Variance Conjectures on the Critical Points of Polynomials. In: Brändén, P., Passare, M., Putinar, M. (eds) Notions of Positivity and the Geometry of Polynomials. Trends in Mathematics. Springer, Basel. https://doi.org/10.1007/978-3-0348-0142-3_16

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