Computing submatrices of the Hermite normal form of a structured polynomial matrix

TL;DR

This paper presents a method to efficiently compute specific submatrices of the Hermite normal form of structured polynomial matrices by exploiting displacement structure and structured linear algebra techniques.

Contribution

It introduces an approach that accelerates the computation of HNF submatrices for structured polynomial matrices using evaluation-interpolation and inverse matrix row recovery.

Findings

Accelerates HNF submatrix computation for structured matrices

Uses evaluation-interpolation to exploit displacement structure

Recovers HNF submatrices efficiently from inverse matrix rows

Abstract

Following several decades of successive algorithmic improvements, works from the 2010s have showed how to compute the Hermite normal form (HNF) of a univariate polynomial matrix within a complexity bound which is essentially that of polynomial matrix multiplication. Recently, several results on bivariate polynomials and Gr\"obner bases have highlighted the interest of computing determinants or HNFs of polynomial matrices that happen to be structured, with a small displacement rank. In such contexts, a small leading principal submatrix of the HNF often contains all the sought information. In this article, we show how the displacement structure can be exploited in order to accelerate the computation of such submatrices. To achieve this, we rely on structured linear algebra over the field thanks to evaluation-interpolation. This allows us to recover some rows of the inverse of the input matrix, from which we deduce the sought HNF submatrix via bases of relations.