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

TL;DR

This paper introduces a generalized Hurwitz matrix and Euclidean algorithm to characterize polynomials with roots outside specific sectors of the complex plane, extending classical stability and root location theorems.

Contribution

It generalizes classical theorems on polynomial stability and root locations by defining a new matrix and algorithm framework for positive coefficient polynomials.

Findings

Polynomials with totally nonnegative generalized Hurwitz matrices do not vanish in certain complex sectors.

The work extends Hurwitz's stability theorem and related classical results.

A new generalized Euclidean algorithm is developed for analyzing polynomial roots.

Abstract

Given a polynomial \[ f(x)=a_0x^n+a_1x^{n-1}+\cdots +a_n \] with positive coefficients $a_k$, and a positive integer $M\leq n$, 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 $$ \left\{z\in\mathbb{C}: |\arg (z)| < \frac{\pi}{M}\right\} $$ 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.