Improved Polynomial Remainder Sequences for Ore Polynomials

TL;DR

This paper extends methods for optimizing polynomial remainder sequences to Ore polynomials, reducing coefficient sizes and providing new bounds and insights into the structure of these sequences.

Contribution

It generalizes two improvements of subresultant sequences to Ore polynomials and derives a new bound for minimal coefficient size, offering a new perspective on extraneous factors.

Findings

Derived a new bound for minimal coefficient size in Ore polynomial remainder sequences.

Generalized improvements of subresultant sequences to Ore polynomials.

Provided a new proof for results in the commutative case, clarifying extraneous factors.

Abstract

Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different ways have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size of the coefficients is optimal in the generic case, but when taking the input from applications, the coefficients are often larger than necessary. We generalize two improvements of the subresultant sequence to Ore polynomials and derive a new bound for the minimal coefficient size. Our approach also yields a new proof for the results in the commutative case, providing a new point of view on the origin of the extraneous factors of the coefficients.