Sturm and Sylvester algorithms revisited via tridiagonal determinantal representations

TL;DR

This paper revisits Sturm and Sylvester algorithms, revealing their connection to dual tridiagonal determinantal representations of polynomials, and establishes a relationship between the number of real roots and the signature of these representations.

Contribution

It introduces a novel interpretation of Sturm and Sylvester algorithms through tridiagonal determinantal representations, linking root counting to matrix signatures.

Findings

Sturm and Sylvester algorithms lead to dual tridiagonal determinantal representations.

Number of real roots exceeds the signature of the associated representation.

Provides a new perspective on root counting via matrix signatures.

Abstract

First, we show that Sturm algorithm and Sylvester algorithm, which compute the number of real roots of a given univariate polynomial, lead to two dual tridiagonal determinantal representations of the polynomial. Next, we show that the number of real roots of a polynomial given by a tridiagonal determinantal representation is greater than the signature of this representation.