On the generalised Ritt problem as a computational problem

TL;DR

This paper explores the generalized Ritt problem, establishing its equivalence to zero divisor testing in radical differential ideals and providing algorithms for canonical decompositions and generators.

Contribution

It introduces multiple equivalent formulations of the generalized Ritt problem and develops algorithms for canonical decomposition and generating sets of radical differential ideals.

Findings

Equivalent formulations of the Ritt problem established

Algorithms for canonical decomposition of radical differential ideals

Algorithms for canonical generating sets independent of generators and ranking

Abstract

The Ritt problem asks if there is an algorithm that tells whether one prime differential ideal is contained in another one if both are given by their characteristic sets. We give several equivalent formulations of this problem. In particular, we show that it is equivalent to testing if a differential polynomial is a zero divisor modulo a radical differential ideal. The technique used in the proof of equivalence yields algorithms for computing a canonical decomposition of a radical differential ideal into prime components and a canonical generating set of a radical differential ideal. Both proposed representations of a radical differential ideal are independent of the given set of generators and can be made independent of the ranking.