Submission date: 20 June 2019

Abstract:

A criterion in the spirit of Rosenlicht is given, on the rational function f(x), for when the planar vector field defined by x'=f(x) and y'=xy admits a pair of algebraically independent first integrals over some extension of the base field.

This proceeds from model-theoretic considerations by working in the theory of differentially closed fields of characteristic zero and asking: If D is a strongly minimal set on the affine line that is internal to the constants, when is the pullback of D under the logarithmic derivative itself internal to the constants?

Mathematics Subject Classification: 03C96, 12H05

Keywords and phrases: logarithmic derivative, internality, differentially closed fields

Full text arXiv 1906.08659: pdf, ps.