### Abstract

Let v(s) d denote the set of coefficient vectors of contractive polynomials of degree d with 2s non-real zeros. We prove that v(s) d 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(s) d 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.

Original language | English |
---|---|

Pages (from-to) | 927-949 |

Number of pages | 23 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 66 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2014 |

### Fingerprint

### Keywords

- Polynomials with bounded roots
- Selberg integral

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**On the distribution of polynomials with bounded roots, I. Polynomials with real coefficients.** / Akiyama, Shigeki; Pethő, A.

Research output: Contribution to journal › Article

*Journal of the Mathematical Society of Japan*, vol. 66, no. 3, pp. 927-949. https://doi.org/10.2969/jmsj/06630927

}

TY - JOUR

T1 - On the distribution of polynomials with bounded roots, I. Polynomials with real coefficients

AU - Akiyama, Shigeki

AU - Pethő, A.

PY - 2014

Y1 - 2014

N2 - Let v(s) d denote the set of coefficient vectors of contractive polynomials of degree d with 2s non-real zeros. We prove that v(s) d 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(s) d 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.

AB - Let v(s) d denote the set of coefficient vectors of contractive polynomials of degree d with 2s non-real zeros. We prove that v(s) d 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(s) d 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.

KW - Polynomials with bounded roots

KW - Selberg integral

UR - http://www.scopus.com/inward/record.url?scp=84890423601&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84890423601&partnerID=8YFLogxK

U2 - 10.2969/jmsj/06630927

DO - 10.2969/jmsj/06630927

M3 - Article

AN - SCOPUS:84890423601

VL - 66

SP - 927

EP - 949

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 3

ER -