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Distribution of Zeros for Random Laurent Rational Functions

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Abstract

We study the asymptotic distribution of zeros for the random rational functions that can be viewed as partial sums of a random Laurent series. If this series defines a random analytic function in an annulus A, then the zeros accumulate on the boundary circles of A, being equidistributed in the angular sense, with probability 1. We also show that the equidistribution phenomenon holds if the annulus of convergence degenerates to a circle. Moreover, equidistribution of zeros still persists when the Laurent rational functions diverge everywhere, which is new even in the deterministic case. All results hold under two types of general conditions on random coefficients. The first condition is that the random coefficients are non-trivial i.i.d. random variables with finite \(\log ^+\) moments. The second condition allows random variables that need not be independent or identically distributed, but only requires certain uniform bounds on the tails of their distributions.

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Acknowledgements

Research was partially supported by the National Security Agency (Grant H98230-15-1-0229) and by the American Institute of Mathematics.

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Authors and Affiliations

  1. Department of Mathematics, Oklahoma State University, Stilwater, OK, 74078, USA

    Igor E. Pritsker

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  1. Igor E. Pritsker
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Correspondence to Igor E. Pritsker.

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Communicated by Doron Lubinsky.

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Pritsker, I.E. Distribution of Zeros for Random Laurent Rational Functions. Comput. Methods Funct. Theory 18, 143–157 (2018). https://doi.org/10.1007/s40315-017-0213-3

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