A proof of the main theorem on Bezoutians

TL;DR

This paper provides a self-contained proof that the nullity of the Bezoutian matrix for two polynomials equals the number of their common zeros, including multiplicities.

Contribution

It offers a new, self-contained proof of a fundamental theorem relating Bezoutian nullity to common zeros of polynomials.

Findings

Nullity of Bezoutian equals the number of common zeros with multiplicities

Provides a self-contained proof of the main theorem

Clarifies the relationship between Bezoutians and polynomial roots

Abstract

We give a self-contained proof that the nullity of the Bezoutian matrix associated with a pair of polynomials $f$ and $g$ equals the number of their common zeros counting multiplicities.