Order one differential equations on nonisotrivial algebraic curves

TL;DR

This paper introduces new examples of strongly minimal differential algebraic varieties on nonisotrivial curves, using a novel theory of τ-forms and connections to deformation theory, advancing the understanding of differential equations on algebraic curves.

Contribution

It develops a theory of τ-forms and applies it to construct new strongly minimal varieties on nonisotrivial curves, answering open questions in the field.

Findings

Constructed new strongly minimal differential varieties on nonisotrivial curves.

Developed a theory of τ-forms linked to deformation theory.

Extended previous work of Buium and Rosen.

Abstract

In this paper we provide new examples of geometrically trivial strongly minimal differential algebraic varieties living on nonisotrivial curves over differentially closed fields of characteristic zero. These are systems whose solutions only have binary algebraic relations between them. Our technique involves developing a theory of $\tau$-forms, and building connections to deformation theory. This builds on previous work of Buium and Rosen. In our development, we answer several open questions posed by Rosen and Hrushovski-Itai.