Remarks on circulant matrices and critical points of polynomials (Q2033174)

The authors study relations between the zeros of a polynomial and the zeros of its derivative, i.e., its {critical points}. They are specifically interested in the ``Schoenberg conjecture'' (see [\textit{I. J. Schoenberg}, Am. Math. Mon. 93, 8--13 (1986; Zbl 0627.30001)]) and its generalizations in [\textit{M. G. de Bruin} and \textit{A. Sharma}, J. Comput. Appl. Math. 105, No. 1--2, 221--228 (1999; Zbl 0945.30003)]. An important role in this matter is played by the connection between critical points and circular matrices, as well as an estimate for the eigenvales of the involved matrices. Let \(z_1,\ldots,z_n\) be the zeros of the polynomial \(p(z)\) and let \(w_1,\ldots,w_{n-1}\) be the critical points of \(p(z)\). Schoenberg's conjecture states that \[\sum_{k=1}^{n-1}\,|w_k|^2\leq\,\frac{1}{n^2}\,\left|\sum_{j=1}^n\,z_j\right|^2+\,\frac{n-2}{n}\,\sum_{j=1}^n\,|z_j|^2,\] while its generalization claims that \[\sum_{k=1}^{n-1}\,|w_k|^4\leq\,\frac{2}{n^2}\,\left(\sum_{j=1}^n\,|z_j|^2\right)^2+\,\frac{n-4}{n}\,\sum_{j=1}^n\,|z_j|^4.\] The new result proved in this paper is the following Theorem. For \(p\geq 2\) there holds \[\sum_{k=1}^{n-1}\,|w_k|^p\leq\,\frac{(n-1)^{p-2}}{n^p}\,\left|\sum_{j=1}^n\,z_j\right|^p+\,\frac{(n-1)^{p-2}(n-2)}{n^{p/2}}\,\left(\sum_{j=1}^n\, |z_j|^2\right)^{p/2}.\]