Fast Computation of Common Left Multiples of Linear Ordinary Differential Operators

TL;DR

This paper introduces a new, nearly optimal algorithm for computing the least common left multiples of linear differential operators, significantly improving efficiency and bounds over previous methods.

Contribution

The paper presents a novel linear algebra-based algorithm for LCLMs that achieves sharper bounds and near-optimal complexity, advancing computational methods in differential algebra.

Findings

Sharp bounds on coefficient degrees of LCLMs

Nearly optimal arithmetic complexity of the new algorithm

Improved efficiency over previous algorithms

Abstract

We study tight bounds and fast algorithms for LCLMs of several linear differential operators with polynomial coefficients. We analyze the arithmetic complexity of existing algorithms for LCLMs, as well as the size of their outputs. We propose a new algorithm that recasts the LCLM computation in a linear algebra problem on a polynomial matrix. This algorithm yields sharp bounds on the coefficient degrees of the LCLM, improving by one order of magnitude the best bounds obtained using previous algorithms. The complexity of the new algorithm is almost optimal, in the sense that it nearly matches the arithmetic size of the output.