On the density of strongly minimal algebraic vector fields

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 $d\ge 2$ on the affine space of dimension $n\ge 2$ 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 Painlevé and satisfy autonomous differential equations of order $n\ge 4$.

arXiv comment58 pages, presentation improved

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• When any three solutions are independent
• Generic planar algebraic vector fields are disintegrated
• Generic differential equations are strongly minimal
• Algebraic relations between solutions of Painlevé equations
• Abelian reduction in differential-algebraic and bimeromorphic geometry
• A differential analogue of the wild automorphism conjecture