Generalized Hurwitz Matrices, Generalized Euclidean Algorithm, and Forbidden Sectors of the Complex Plane

Abstract

Given a polynomial

$$\begin{aligned} f(x)=a_0x^n+a_1x^{n-1}+\cdots +a_n \end{aligned}$$

with positive coefficients $a_k$, and a positive integer $M\le n$, we define an 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

$$\begin{aligned} \left\{ z\in \mathbb {C}: |\arg (z)| < \frac{\pi }{M}\right\} \end{aligned}$$

whenever the matrix $H_M$ is totally non-negative. 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.