Asymptotically fast polynomial matrix algorithms for multivariable systems

TL;DR

This paper introduces the fastest algorithms for key polynomial matrix problems, reducing their complexity to that of polynomial matrix multiplication, thus significantly improving computational efficiency.

Contribution

It demonstrates that fundamental polynomial matrix problems can be solved in asymptotically optimal time by reducing them to minimal basis computations and matrix multiplication.

Findings

Algorithms for rank, nullspace, determinant, inverse, and reduced form are asymptotically fastest.

All problems can be solved in about the same time as polynomial matrix multiplication.

Reductions rely on minimal basis computations and matrix fraction techniques.

Abstract

We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form. We show that they essentially can be reduced to two computer algebra techniques, minimal basis computations and matrix fraction expansion/reconstruction, and to polynomial matrix multiplication. Such reductions eventually imply that all these problems can be solved in about the same amount of time as polynomial matrix multiplication.