Generic planar algebraic vector fields are disintegrated

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 set of solutions of a general differential equation of order $2$ and of degree $d\ge 3$ in a differentially closed field is strongly minimal and disintegrated. We also give two other formulations of this result in terms of algebraic (non)-integrability and algebraic independence of the analytic solutions of a general planar algebraic vector field.

27 pages

References in corpus (3)

• On algebraic relations between solutions of a generic Painleve equation
• Rational factors, invariant foliations and algebraic disintegration of compact mixing Anosov flow of dimension $3$
• Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups

Cited by in corpus (2)

• When any three solutions are independent
• On the density of strongly minimal algebraic vector fields