A fast, deterministic algorithm for computing a Hermite Normal Form of a polynomial matrix

TL;DR

This paper presents a fast, deterministic algorithm for computing the Hermite normal form of a polynomial matrix with improved complexity, leveraging advanced matrix multiplication techniques.

Contribution

It introduces a novel, efficient method for computing the Hermite normal form of polynomial matrices with complexity depending on matrix multiplication exponent.

Findings

Algorithm achieves complexity $O^{ ilde}(n^{\omega}d)$ for Hermite form computation.

Uses a fast method to determine diagonal entries with complexity $O^{ ilde}(n^{\omega}s)$.

Provides a deterministic approach with improved efficiency over previous methods.

Abstract

Given a square, nonsingular matrix of univariate polynomials $\mathbf{F} \in \mathbb{K}[x]^{n \times n}$ over a field $\mathbb{K}$, we give a fast, deterministic algorithm for finding the Hermite normal form of $\mathbf{F}$ with complexity $O^{\sim}\left(n^{\omega}d\right)$ where $d$ is the degree of $\mathbf{F}$. Here soft-$O$ notation is Big-$O$ with log factors removed and $\omega$ is the exponent of matrix multiplication. The method relies of a fast algorithm for determining the diagonal entries of its Hermite normal form, having as cost $O^{\sim}\left(n^{\omega}s\right)$ operations with $s$ the average of the column degrees of $\mathbf{F}$.