Journal of the Mathematical Society of Japan

On the distribution of polynomials with bounded roots, I. Polynomials with real coefficients

Shigeki AKIYAMA and Attila PETHŐ

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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.

Article information

Source
J. Math. Soc. Japan, Volume 66, Number 3 (2014), 927-949.

Dates
First available in Project Euclid: 24 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1406206977

Digital Object Identifier
doi:10.2969/jmsj/06630927

Mathematical Reviews number (MathSciNet)
MR3238322

Zentralblatt MATH identifier
1301.53040

Subjects
Primary: 33B20: Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)
Secondary: 11B65: Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30] 93D10: Popov-type stability of feedback systems

Keywords
Selberg integral polynomials with bounded roots

Citation

AKIYAMA, Shigeki; PETHŐ, Attila. On the distribution of polynomials with bounded roots, I. Polynomials with real coefficients. J. Math. Soc. Japan 66 (2014), no. 3, 927--949. doi:10.2969/jmsj/06630927. https://projecteuclid.org/euclid.jmsj/1406206977


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