Simple algorithm for GCD of polynomials

TL;DR

This paper introduces a straightforward, division-free algorithm for computing the GCD of two polynomials based on the Bezout approach, which also allows calculation of discriminants and resultants efficiently.

Contribution

It presents a novel, simple algorithm that avoids division and determinants, reducing complexity in polynomial GCD computations and related discriminant and resultant calculations.

Findings

Requires only n steps for degree n polynomials

Eliminates need for division or determinant calculations

Enables direct computation of discriminants and resultants

Abstract

Based on the Bezout approach we propose a simple algorithm to determine the {\tt gcd} of two polynomials which doesn't need division, like the Euclidean algorithm, or determinant calculations, like the Sylvester matrix algorithm. The algorithm needs only $n$ steps for polynomials of degree $n$. Formal manipulations give the discriminant or the resultant for any degree without needing division nor determinant calculation.