Effective definability of Kolchin polynomials

TL;DR

This paper demonstrates that the Kolchin polynomial, a differential-algebraic rank in differential algebraic geometry, is definable in families, unlike other model-theoretic ranks, and also shows the definability of weak irreducibility.

Contribution

It proves the definability of the Kolchin polynomial and weak irreducibility in families of differential varieties, advancing understanding of definability in differential algebraic geometry.

Findings

Kolchin polynomial is definable in families

Weak irreducibility is definable in families

Full irreducibility remains an open problem

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.