Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix

TL;DR

This paper presents fast, deterministic algorithms for computing the determinant and Hermite normal form of nonsingular polynomial matrices, improving efficiency using advanced matrix triangularization techniques.

Contribution

It introduces novel deterministic algorithms with optimal complexity for polynomial matrix Hermite form and determinant computation, based on a new triangularization method.

Findings

Algorithms run in ( abla n^\u03a9 s) operations

Deterministic approach improves reliability over probabilistic methods

Complexity depends on matrix size and degree bounds

Abstract

Given a nonsingular $n \times n$ matrix of univariate polynomials over a field $\mathbb{K}$, we give fast and deterministic algorithms to compute its determinant and its Hermite normal form. Our algorithms use $\widetilde{\mathcal{O}}(n^\omega \lceil s \rceil)$ operations in $\mathbb{K}$, where $s$ is bounded from above by both the average of the degrees of the rows and that of the columns of the matrix and $\omega$ is the exponent of matrix multiplication. The soft-$O$ notation indicates that logarithmic factors in the big-$O$ are omitted while the ceiling function indicates that the cost is $\widetilde{\mathcal{O}}(n^\omega)$ when $s = o(1)$. Our algorithms are based on a fast and deterministic triangularization method for computing the diagonal entries of the Hermite form of a nonsingular matrix.