The Distribution of Zeros of the Derivative of a Random Polynomial

Robin Pemantle () and Igor Rivin ()
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Robin Pemantle: University of Pennsylvania, Department of Mathematics
Igor Rivin: Temple University, Department of Mathematics

A chapter in Advances in Combinatorics, 2013, pp 259-273 from Springer

Abstract: Abstract In this note we initiate the probabilistic study of the critical points of polynomials of large degree with a given distribution of roots. Namely, let f be a polynomial of degree n whose zeros are chosen IID from a probability measure μ on $$\mathbb{C}$$ . We conjecture that the zero set of f ′ always converges in distribution to μ as n → ∞. We prove this for measures with finite one-dimensional energy. When μ is uniform on the unit circle this condition fails. In this special case the zero set of f ′ converges in distribution to that of the IID Gaussian random power series, a well known determinantal point process.

Keywords: Gauss-Lucas theorem; Gaussian series; Critical points; Random polynomials (search for similar items in EconPapers)
Date: 2013
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-30979-3_14

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DOI: 10.1007/978-3-642-30979-3_14

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