Computing the Hermite Form of a Matrix of Ore Polynomials

TL;DR

This paper presents an efficient algorithm for computing the Hermite form of matrices over Ore polynomial rings, with polynomial complexity in dimensions and degrees, applicable to differential and shift polynomial cases.

Contribution

It introduces a polynomial-time algorithm for Hermite form computation over Ore polynomial rings, utilizing Dieudonne determinants and quasideterminants for explicit bounds.

Findings

Algorithm runs in polynomial time in matrix size and degree.

Explicit complexity bounds for differential and shift polynomial cases.

Applicable to fields like Q(z) and Q with efficient performance.

Abstract

Let R=F[D;sigma,delta] be the ring of Ore polynomials over a field (or skew field) F, where sigma is a automorphism of F and delta is a sigma-derivation. Given a an m by n matrix A over R, we show how to compute the Hermite form H of A and a unimodular matrix U such that UA=H. The algorithm requires a polynomial number of operations in F in terms of both the dimensions m and n, and the degree of the entries in A. When F=k(z) for some field k, it also requires time polynomial in the degree in z, and if k is the rational numbers Q, it requires time polynomial in the bit length of the coefficients as well. Explicit analyses are provided for the complexity, in particular for the important cases of differential and shift polynomials over Q(z). To accomplish our algorithm, we apply the Dieudonne determinant and quasideterminant theory for Ore polynomial rings to get explicit bounds on the degrees and sizes of entries in H and U.