Computation of gcd chain over the power of an irreducible polynomial

TL;DR

This paper presents a detailed algorithm for computing gcd chains over powers of irreducible polynomials, a special case that sheds light on the general problem of gcd computation in polynomial rings modulo primary triangular sets.

Contribution

It provides a complete algorithm for the specific case of a power of an irreducible polynomial, illustrating key steps for the broader problem of gcd chain computation.

Findings

The algorithm effectively computes gcd chains in the simplified case.

Key steps identified for extending to more complex triangular sets.

Insights into the challenges of generalizing gcd chain algorithms.

Abstract

A notion of gcd chain has been introduced by the author at ISSAC 2017 for two univariate monic polynomials with coefficients in a ring R = k[x_1, ..., x_n ]/(T) where T is a primary triangular set of dimension zero. A complete algorithm to compute such a gcd chain remains challenging. This work treats completely the case of a triangular set T = (T_1 (x)) in one variable, namely a power of an irreducible polynomial. This seemingly "easy" case reveals the main steps necessary for treating the general case, and it allows to isolate the particular one step that does not directly extend and requires more care.