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On the distribution of polynomials with bounded roots. I: Polynomials with real coefficients. (English) Zbl 1305.33006

Let \(d\) be a positive integer, and denote by \(E_d\) the set of all \(d\)-dimensional polynomials whose roots lie within the ball of radius 1 centered at the origin. This set was first studied by I. Schur in 1918, who found conditions, which imply that the boundary of \(E_d\) is the union of finitely many algebraic surfaces. In [IEEE Trans. Autom. Control 23, 454–458 (1978; Zbl 0377.93021)] A. T. Fam and J. S. Meditch improved this result by showing that the boundary of \(E_d\) is the union of two hyperplanes and one hypersurfaces.
The authors of the paper under review prove a generalization of the above results for the boundary of the set \(v^{(s)}_d\), which is the set of polynomials having signature \(s\). They prove that the \(d\)-dimensional Lebesgue measure of this last set can be computed by a certain multiple integral (which is too complicated to be stated here), related to the well-known Selberg integral and its generalization.

MSC:

11C08 Polynomials in number theory
11B65 Binomial coefficients; factorials; \(q\)-identities
26B15 Integration of real functions of several variables: length, area, volume
26C05 Real polynomials: analytic properties, etc.
33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)

Citations:

Zbl 0377.93021
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References:

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