Generic planar algebraic vector fields are strongly minimal and disintegrated

Jaoui, Rémi

doi:10.2140/ant.2021.15.2449

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Generic planar algebraic vector fields are strongly minimal and disintegrated

Rémi Jaoui

Vol. 15 (2021), No. 10, 2449–2483

DOI: 10.2140/ant.2021.15.2449

Abstract

We study model-theoretic properties of algebraic differential equations of order two defined over constant differential fields. In particular, we show that the set of solutions of a “general” differential equation of order two and of degree $d \geq 3$ in a differentially closed field is strongly minimal and disintegrated (in other words, is strongly minimal with trivial forking geometry).

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.

Keywords

model theory, differential algebra, strong minimality, foliations and webs

Mathematical Subject Classification

Primary: 03C60

Secondary: 12H05

Milestones

Received: 20 November 2019

Revised: 3 February 2021

Accepted: 15 April 2021

Published: 8 February 2022

Authors

Rémi Jaoui
Department of Mathematics University of Notre Dame South Bend, IN United States

Vol. 15, No. 10, 2021