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 talk we show that the differential-algebraic rank given by the Kolchin polynomial is in fact definable. This result relies on a uniform bound on the Hilbert-Kolchin index. As a byproduct, we show 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.