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

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 $O^{\sim }\left(n^{\ddot{I}}‰d\right)$ where $d$ is the degree of $\mathbf{F}$. Here soft-$O$ notation is Big-$O$ with log factors removed and $\ddot{I}‰$ 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 $O^{\sim }\left(n^{\ddot{I}}‰s\right)$ operations with $s$ the average of the column degrees of $\mathbf{F}$.