Generalized Hurwitz matrices, generalized Euclidean algorithm, and forbidden sectors of the complex plane

Given a polynomial $$f(x)=a_0{x}^n+a_1{x}^{n-1}+\cdots +a_n$$ with positive coefficients $a_k$, and a positive integer $M\le n$, we define a(n infinite) generalized Hurwitz matrix $H_M(f):=(a_{Mj-i}{)}_{i,j}$. We prove that the polynomial $f(z)$ does not vanish in the sector $$\{z\in \mathbb{C}:\mathrm{\mid }\arg (z)\mathrm{\mid }<\frac{\pi }{M}\}$$ whenever the matrix $H_M$ is totally nonnegative. This result generalizes the classical Hurwitz’ Theorem on stable polynomials ($M=2$), the Aissen-Edrei-Schoenberg-Whitney theorem on polynomials with negative real roots ($M=1$), and the Cowling-Thron theorem ($M=n$). In this connection, we also develop a generalization of the classical Euclidean algorithm, of independent interest per se.