On the distribution of polynomials with bounded roots. I: Polynomials with real coefficients (Q743696)

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 \textit{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)] \textit{A. T. Fam} and \textit{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.