On the generalized Ritt problem as a computational problem

Abstract

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.

Proof mining and effective bounds in differential polynomial rings

“…(In the differential setting, subscripts now indicate bounds on order as well as degree; see Definition 5.2.) Moreover, Corollary 5.50 and Lemma 7.21 give an explicit bound, not on primality of differential ideals (which is open and equivalent to the well-known Ritt problem [19]), but on the weaker Theorem 5.4 of [25] that bounds only one factor:…”

Effective definability of Kolchin polynomials

“…Note that being weakly irreducible is not equivalent to being (fully) irreducible. In fact the question of definability of irreducibility for differential varieties is remarkably difficult (and remains open); it is actually equivalent to the generalized Ritt problem which is a longstanding problem since the 1950's (see [4,Theorem 1]).…”

On the model companion of partial differential fields with an automorphism

“…is definable (see [10] and [13] for partial results around this problem). The other problem is that it is not known if differential dominance onto irreducible ∆-closed sets is a definable condition.…”