Fast and deterministic computation of the determinant of a polynomial matrix

TL;DR

This paper presents a deterministic algorithm for efficiently computing the determinant of a polynomial matrix with complexity depending on matrix size and average degree, improving computational speed and reliability.

Contribution

The authors introduce a novel deterministic method for calculating polynomial matrix determinants with optimal complexity bounds, advancing beyond previous probabilistic approaches.

Findings

Algorithm operates in O(n^ω s) field operations

Deterministic approach ensures reliable results

Applicable to matrices with high degrees and sizes

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 deterministic algorithm for finding the determinant of $\mathbf{F}$. The complexity of the algorithm is $\bigO \left(n^{\omega}s\right)$ field operations where $s$ is the average column degree or the average row degree of $\mathbf{F}$. Here $\bigO$ notation is Big-$O$ with log factors omitted and $\omega$ is the exponent of matrix multiplication.