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July, 2014 On the distribution of polynomials with bounded roots, I. Polynomials with real coefficients
Shigeki AKIYAMA, Attila PETHŐ
J. Math. Soc. Japan 66(3): 927-949 (July, 2014). DOI: 10.2969/jmsj/06630927

Abstract

Let $v_d^{(s)}$ denote the set of coefficient vectors of contractive polynomials of degree $d$ with $2s$ non-real zeros.We prove that $v_d^{(s)}$ can be computed by a multiple integral, which is related to the Selberg integral and its generalizations. We show that the boundary of the above set is the union of finitely many algebraic surfaces. We investigate arithmetical properties of $v_d^{(s)}$ and prove among others that they are rational numbers. We will show that within contractive polynomials, the ‘probability’ of picking a totally real polynomial decreases rapidly when its degree becomes large.

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Shigeki AKIYAMA. Attila PETHŐ. "On the distribution of polynomials with bounded roots, I. Polynomials with real coefficients." J. Math. Soc. Japan 66 (3) 927 - 949, July, 2014. https://doi.org/10.2969/jmsj/06630927

Information

Published: July, 2014
First available in Project Euclid: 24 July 2014

zbMATH: 1301.53040
MathSciNet: MR3238322
Digital Object Identifier: 10.2969/jmsj/06630927

Subjects:
Primary: 33B20
Secondary: 11B65, 93D10

Rights: Copyright © 2014 Mathematical Society of Japan

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Vol.66 • No. 3 • July, 2014
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