Submission date: 6 June 2018

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.

Mathematics Subject Classification: 2H05, 14Q20

Keywords and phrases: differential fields, Kolchin polynomial, effective definability in families

Full text arXiv 1806.02060: pdf, ps.