Computing the Rank and a Small Nullspace Basis of a Polynomial Matrix

TL;DR

This paper presents a randomized algorithm that efficiently computes the rank and a small nullspace basis of a univariate polynomial matrix by reducing the problem to polynomial matrix multiplication, matching the complexity of matrix multiplication.

Contribution

It introduces a novel reduction of the rank and nullspace basis computation to polynomial matrix multiplication, achieving near-optimal complexity with a Las Vegas randomized approach.

Findings

Algorithm matches matrix multiplication complexity in polynomial matrices

Uses matrix Hensel lifting and minimal fraction reconstruction techniques

Provides Las Vegas certification for correctness

Abstract

We reduce the problem of computing the rank and a nullspace basis of a univariate polynomial matrix to polynomial matrix multiplication. For an input n x n matrix of degree d over a field K we give a rank and nullspace algorithm using about the same number of operations as for multiplying two matrices of dimension n and degree d. If the latter multiplication is done in MM(n,d)=softO(n^omega d) operations, with omega the exponent of matrix multiplication over K, then the algorithm uses softO(MM(n,d)) operations in K. The softO notation indicates some missing logarithmic factors. The method is randomized with Las Vegas certification. We achieve our results in part through a combination of matrix Hensel high-order lifting and matrix minimal fraction reconstruction, and through the computation of minimal or small degree vectors in the nullspace seen as a K[x]-module