Sturm's Theorem with Endpoints

TL;DR

This paper offers a concise, direct proof of Sturm's Theorem, addressing the challenging case where an interval endpoint is a root of the polynomial, which is often overlooked.

Contribution

It provides a simplified proof of Sturm's Theorem that explicitly handles the case of roots at interval endpoints, enhancing understanding and correctness.

Findings

A short, direct proof of Sturm's Theorem is presented.

The proof explicitly includes the case where an endpoint is a root.

The approach clarifies the sign alternation method for counting roots.

Abstract

Sturm's Theorem is a fundamental 19th century result relating the number of real roots of a polynomial $f$ in an interval to the number of sign alternations in a sequence of polynomial division-like calculations. We provide a short direct proof of Sturm's Theorem, including the numerically vexing case (ignored in many published accounts) where an interval endpoint is a root of $f$.