Computing Popov and Hermite forms of rectangular polynomial matrices

TL;DR

This paper introduces fast deterministic algorithms for computing Popov and Hermite forms of rectangular polynomial matrices, improving efficiency and matching or surpassing previous randomized methods.

Contribution

It provides the first deterministic algorithms for rectangular matrices, with optimal or improved cost bounds for Popov and Hermite forms.

Findings

Cost bound for Popov form matches best known randomized algorithms.

Cost bound for Hermite form is significantly improved, especially for large matrices.

Algorithms are deterministic and efficient for rectangular polynomial matrices.

Abstract

We consider the computation of two normal forms for matrices over the univariate polynomials: the Popov form and the Hermite form. For matrices which are square and nonsingular, deterministic algorithms with satisfactory cost bounds are known. Here, we present deterministic, fast algorithms for rectangular input matrices. The obtained cost bound for the Popov form matches the previous best known randomized algorithm, while the cost bound for the Hermite form improves on the previous best known ones by a factor which is at least the largest dimension of the input matrix.