Polynomials whose coefficients coincide with their zeros

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Abstract

In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results. We obtain estimates on the number of Ulam polynomials of degree N. We provide additional methods to obtain algebraic identities satisfied by the zeros of Ulam polynomials, beyond the straightforward comparison of their zeros and coefficients. To address the question about the existence of orthogonal Ulam polynomial sequences, we show that the only Ulam polynomial eigenfunctions of hypergeometric type differential operators are the trivial Ulam polynomials \(\{x^N\}_{N=0}^\infty \). We propose a family of solvable N-body problems such that their stable equilibria are the zeros of certain Ulam polynomials.

Keywords

Enumerative geometry Ulam polynomials Ulam map Special polynomials Location of zeros 

Mathematics Subject Classification

26C10 12E10 14N10 33C45 

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Notes

Acknowledgements

The work on this paper began in Summer 2015 during O. Bihun’s visit to the “La Sapienza” University of Rome. The first electronic draft is dated February 11, 2016. Some more work on the paper was done during O. Bihun’s visit to the “La Sapienza” University of Rome in June 2016 and D. Fulghesu’s visit to the Scuola Normale Superiore in Pisa in June–July 2016. Both authors are grateful for the hospitality of the respective institutions. D. Fulghesu’s research is supported in part by the Simons Foundation Collaborations for Mathematicians Grant #360311.

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Colorado, Colorado SpringsColorado SpringsUSA
  2. 2.Department of MathematicsMinnesota State University MoorheadMoorheadUSA

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