On the density of strongly minimal algebraic vector fields

TL;DR

This paper proves that generic algebraic vector fields of degree at least 2 are strongly minimal and geometrically trivial, providing new examples of meromorphic functions satisfying high-order autonomous differential equations.

Contribution

It establishes the abundance of strongly minimal, geometrically trivial differential equations for generic algebraic vector fields and meromorphic functions of high order.

Findings

Generic degree ≥ 2 algebraic vector fields are strongly minimal.

High-order differential equations have new meromorphic solutions.

Examples of autonomous differential equations of order ≥ 4 are constructed.

Abstract

Two theorems witnessing the abundance of geometrically trivial strongly minimal autonomous differential equations of arbitrary order are shown. The first one states that a generic algebraic vector field of degree $d\geq 2$ on the affine space of dimension $n \geq 2$ is strongly minimal and geometrically trivial. The second one states that if $X_0$ is the complement of a smooth hyperplane section $H$ of a smooth projective variety $X$ of dimension $n$, then for $d$ large enough, the system of differential equations associated with a generic vector field on $X_0$ with a pole of order at most $d$ along $H$ is strongly minimal and geometrically trivial. This produces the first examples of meromorphic functions that are new in the sense of Painlev\'e and satisfy autonomous differential equations of order $n \geq 4$.