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 + ⋯ + a n with positive coefficients a k , and a positive integer M ≤ n , we define an infinite generalized Hurwitz matrix H M ( f ) : = ( a M j - i ) i , j . We prove that the polynomial f ( z ) does not vanish in the sector z ∈ C : | arg ( z ) | < π M 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.