An inequality about the largest roots of a polynomial (Q1392666)

The first results giving bounds for the roots of complex polynomials go back to \textit{Cauchy} (1829). The problem considered in this paper, to bound the product of the \(k\) largest roots of a polynomial of degree \(n\) with complex coefficients, was first considered by \textit{W. Specht} [Jahresber. Dtsch. Math.-Ver. 48, Abt. 1, 142-145 (1938; Zbl 0022.30102)], who used compound matrices related to the polynomial. There was another result obtained much latter by \textit{M. Mignotte} [Elem. Math. 46, 85-86 (1991; Zbl 0745.12001)], using elementary algebra, but this result is rarely better than Specht's. Here the author uses again compound matrices, but studies these matrices in great detail, using \textit{Mathematica} to make some symbolic computation. In this way he is able to get almost the best possible results given by this method and thus to improve on Specht's estimates.