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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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

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

First available in Project Euclid: 24 July 2014

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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

Selberg integral polynomials with bounded roots


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.

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  • S. Akiyama and A. Pethő, On the distribution of polynomials with bounded roots II. Polynomials with integer coefficients, submitted.
  • G. W. Anderson, A short proof of Selberg's generalized beta formula, Forum Math., 3 (1991), 415–417.
  • G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia Math. Appl., 71, Cambridge University Press, 1999.
  • K. Aomoto, Configurations and invariant theory of Gauss-Manin systems, In: Group Representations and Systems of Differential Equations, University of Tokyo, 1982, (ed. K. Okamoto), Adv. Stud. Pure Math., 4, Kinokuniya, Tokyo, North-Holland, Amsterdam, 1984, pp.,165–179.
  • K. Aomoto, Jacobi polynomials associated with Selberg integrals, SIAM J. Math. Anal., 18 (1987), 545–549.
  • A. T. Fam and J. S. Meditch, A canonical parameter space for linear systems design, IEEE Trans. Automat. Control, 23 (1978), 454–458.
  • A. Fam, The volume of the coefficient space stability domain of monic polynomials, In: 1989 IEEE International Symposium on Circuits and Systems, 3, may 1989, pp.,1780–1783.
  • Peter J. Forrester and S. Ole Warnaar, The importance of the Selberg integral, Bull. Amer. Math. Soc. (N.S.), 45 (2008), 489–534.
  • A. Granville, Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers, In: Organic Mathematics, Burnaby, BC, 1995, (eds. J. Borwein, P. Borwein, L. Jörgenson and R. Corless), CMS Conf. Proc., 20, Amer. Math. Soc., Providence, RI, 1997, pp.,253–276.
  • P. Kirschenhofer, A. Pethő, P. Surer and J. Thuswaldner, Finite and periodic orbits of shift radix systems, J. Théor. Nombres Bordeaux, 22 (2010), 421–448.
  • E. E. Kummer, Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. Reine Angew. Math., 44 (1852), 93–146; Collected papers, Springer-Verlag, Berlin, 1975.
  • I. Schur, Über Potenzreihen, die im Inneren des Einheitskreises beschränkt sind II, J. Reine Angew. Math., 148 (1918), 122–145.\footnote[3]Reprinted in Schur's collected papers: I. Schur, Gesammelte Abhandlungen. Band I.–III. (German) Herausgegeben von Alfred Brauer und Hans Rohrbach. Springer-Verlag, Berlin-New York, 1973.
  • A. Selberg, Bemerkninger om et multipelt integral, Norsk. Mat. Tidsskr., 24 (1944), 71–78.