In minisymposium: Total Positivity
Tue 16:00–16:30, Auditorium Max WeberFor a given real polynomial % $$ p(z)=a_0z^n+a_1z^{n-1}+\cdots+a_n, \qquad a_0>0, $$ % and an integer $M$, $2\leqslant M\leqslant n$, the infinite matrix $H^{(M)}_{\infty}=(a_{Mj-i})_{i,j\in\mathbb{Z}}$ is called generalized Hurwitz matrix, and for $M=2$ the matrix $H^{(2)}_{\infty}$ is the standard infinite Hurwitz matrix. It is known [1,3] that the total positivity of the matrix $H^{(2)}_{\infty}$ is equivalent to the positivity of the first $n$ leading principal minors of $H^{(2)}_{\infty}$. In [2] there were found finitely many sufficient conditions for the matrix $H^{(M)}_{\infty}$ to be totally positive. In this talk, we show that positivity of finitely many certain minors of the generalized Hurwitz matrix $H^{(M)}_{\infty}$, $2\leqslant M\leqslant n$, is necessary and sufficient for total positivity of the matrix $H^{(M)}_{\infty}$.
We also present some applications of totally positive generalized Hurwitz matrices to the root location of polynomials.
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