Vol. 15, No. 10, 2021

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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 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