Dimension Polynomials of Intermediate Fields and Krull-type Dimension of Finitely Generated Differential Field Extensions

Let K be a differential field of zero characteristic and let L be a finitely generated differential field extension of K. We introduce the concept of a transcendence type and dimension of the extension L/K considering chains of intermediate differentials fields of this extension. Using the technique of differential dimension polynomials we obtain relationships between the transcendence type and dimension of L/K and differential birational invariants of this extension carried by its dimension polynomials.