Submission date: 21 May 2019

Abstract:

In this article, we study model-theoretic properties of algebraic differential equations of order 2, defined over constant differential fields. In particular, we show that the existentially closed theory associated to a general differential equation of order 2 and of degree d ≥ 3 is almost strongly minimal and disintegrated. We also formulate - in the language of algebraic varieties endowed with vector fields - a geometric counterpart of this model-theoretic result. These results provide a positive answer, concerning planar vector fields, to a conjecture of Poizat asserting that (the generic type of) a general differential equation of order n ≥ 2 is always disintegrated.

Mathematics Subject Classification: 12H05, 03C98, 34M15

Keywords and phrases:

Full text arXiv 1905.09429: pdf, ps.