Effective definability of Kolchin polynomials

Abstract

While the natural model-theoretic ranks available in differentially closed fields (of characteristic zero), namely Lascar and Morley rank, are known not to be definable in families of differential varieties; in this note we show that the differential-algebraic rank given by the Kolchin polynomial is in fact definable. As a byproduct, we are able to prove that the property of being weakly irreducible for a differential variety is also definable in families. The question of full irreducibility remains open, it is known to be equivalent to the generalized Ritt problem.

Power Series Representations of Hypergeometric Type Functions

“…For simplification purposes, we introduce a derivation rule that will enable us to unify all desired concepts algorithmically, though explicit links to this derivation can be avoided without great difficulty. In [15] and many references in differential algebra (see also [7]), the notion of order for partial differential rings is different from our use. The prevalent definition does not easily adapt to our computational goal.…”