Geodesic
A simple proof of Hermite's theorem on the zeros of a polynomial
Glasgow mathematical journal, Tome 24 (1983) no. 2, pp. 125-128

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Young, N. J. A simple proof of Hermite's theorem on the zeros of a polynomial. Glasgow mathematical journal, Tome 24 (1983) no. 2, pp. 125-128. doi: 10.1017/S0017089500005176
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[1] 1.Hermite, C., Sur le nombre des racines d'une équation algébrique comprises entre des limites données, J. Reine Angew. Math. 52 (1856), 39–51; translation in Internal. J. Control 26 (1977), 183–195. Google Scholar

[2] 2.Parks, P. C., Further comments on “A symmetric matrix formulation of the Hurwitz-Routh stability criterion”, IEEE Trans. Automat. Control 8 (1963), 270–271. Google Scholar

[3] 3.Parks, P. C., A new proof of Hermite's stability criterion and a generalization of Orlando's formula, Intemat. J. Control 26 (1977), 197–206. Google Scholar

[4] 4.Pták, Vlastimil and Young, N. J., A generalization of the zero location theorem of Schur and Cohn, IEEE Trans. Automat. Control 25 (1980), 978–980. Google Scholar

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