The Hermite-Sylvester criterion for real-rooted polynomials

TL;DR

This paper provides a straightforward proof of the Hermite-Sylvester theorem, establishing that a polynomial is real-rooted if and only if a related quadratic form is positive semidefinite, thus linking root properties to quadratic forms.

Contribution

It offers a simple proof of the Hermite-Sylvester criterion connecting real-rootedness of polynomials with positive semidefiniteness of quadratic forms.

Findings

Proof simplifies understanding of the Hermite-Sylvester theorem

Establishes a clear criterion for real-rootedness via quadratic forms

Enhances theoretical tools for analyzing polynomial roots

Abstract

A polynomial is real-rooted if all of its roots are real. This note gives a simple proof of the Hermite-Sylvester theorem that a polynomial $f(x) \in {\mathbf R}[x]$ is real-rooted if and only if an associated quadratic form is positive semidefinite.