On a theorem of Hermite and Hurwitz

Abstract

The Hermite-Hurwitz theorem computes the degree, over R , of a real rational function ƒ in terms of the signature of an associated quadratic form—known today as the Hankel matrix of ƒ. This formula, which Hermite was led to by his work on the problem of representing integers as sums of squares, gave rise to striking applications in the theory of equations and in the stability theory of ordinary differential equations. In this paper, this theorem and various generalizations to the matrix-valued case are discussed and described in terms of signature formulae. These include its relation to stability theory and the matrix Hermite-Hurwitz theorem of Bitmead-Anderson as applied to questions of circuit synthesis. This also includes a global form of Hörmander's signature formula for the Maslov index of a rational loop in a Lagrangian Grassmannian, due to Byrnes and Duncan, and applications to the topology of spaces of rational matrix-valued functions, following the work of Brockett, Byrnes, and Duncan. This includes, in particular, a topological proof of the matrix Hermite-Hurwitz theorem.