A simple proof of Hermite's theorem on the zeros of a polynomial

Extract

In 1856 Hermite showed how to determine by purely rational operations the number of zeros of a given polynomial lying in a specified half plane [1]: one inspects the signature of a certain Hermitian form. This type of result is still of interest for practical applications, and several authors have provided alternatives to Hermite's original, highly computational proof (for example [2, 3]). Recently V. Pták and the author gave a simple matrix-theoretic proof and generalization of a class of Hermite-type theorems [4]. This class included the Schur-Cohn test for zeros in a circle, but not, to our regret, the original theorem of Hermite. The purpose of this note is to show that a slight modification of our method does indeed provide a simple proof of Hermite's theorem.