2023/01/16 by Rémi Jaoui, Jaoui, Rémi · 1 citation
Mathematics · #03C45 #12H05 #32M25 #34M45 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Logic (math.LO) #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2301.06362
openalex publication_date 2023/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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≥ 2 on the affine space of dimension n ≥ 2 is strongly minimal and geometrically trivial. The second one states that if X0 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 X0 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 ≥ 4.