On computing the Hermite form of a matrix of differential polynomials

TL;DR

This paper presents an efficient algorithm for computing the Hermite form of matrices over differential polynomial rings, with polynomial complexity in matrix size, degrees, and coefficient size, applicable over rational numbers.

Contribution

It introduces a novel polynomial-time algorithm for Hermite form computation of differential polynomial matrices, extending existing methods to this algebraic setting.

Findings

Algorithm runs in polynomial time relative to matrix size and degrees

Applicable over fields of rational numbers with polynomial bit-length complexity

Provides a constructive method for Hermite form and unimodular transformation

Abstract

Given an n x n matrix over the ring of differential polynomials F(t)[\D;\delta], 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 terms of n, deg_D(A), and deg_t(A). When F is the field of rational numbers, it also requires time polynomial in the bit-length of the coefficients.