Rank-Sensitive Computation of the Rank Profile of a Polynomial Matrix

TL;DR

This paper introduces efficient algorithms for computing the column rank profile of polynomial matrices, improving existing methods with rank-sensitive complexity and practical performance benefits.

Contribution

It presents two algorithms for the column rank profile of polynomial matrices, with one offering a rank-sensitive complexity improvement over previous algorithms.

Findings

Improved minimal kernel basis algorithm for polynomial matrices.

A new rank-sensitive algorithm with complexity $O ilde{~}(r^{ ext{}\omega-2} n (m+D))$ operations.

Enhanced computational efficiency for rank profile determination.

Abstract

Consider a matrix $\mathbf{F} \in \mathbb{K}[x]^{m \times n}$ of univariate polynomials over a field $\mathbb{K}$. We study the problem of computing the column rank profile of $\mathbf{F}$. To this end we first give an algorithm which improves the minimal kernel basis algorithm of Zhou, Labahn, and Storjohann (Proceedings ISSAC 2012). We then provide a second algorithm which computes the column rank profile of $\mathbf{F}$ with a rank-sensitive complexity of $O\tilde{~}(r^{\omega-2} n (m+D))$ operations in $\mathbb{K}$. Here, $D$ is the sum of row degrees of $\mathbf{F}$, $\omega$ is the exponent of matrix multiplication, and $O\tilde{~}(\cdot)$ hides logarithmic factors.