2021/01/21 by César Camacho, Camacho, César, Hossein Movasati +1 · 1 citation
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Complex Variables (math.CV) #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2101.08627
openalex publication_date 2021/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a field k of characteristic 0, not necessarily algebraically closed, and a fixed algebraic curve f=0 defined by a tame polynomial f∈ k[x,y] with only quasi-homogeneous singularities. We prove that the space of holomorphic foliations in the plane mathbb A2_\k having f=0 as a fixed invariant curve is generated as k[x,y]-module by at most four elements, three of them are the trivial foliations fdx,fdy and df. Our proof is algorithmic and constructs the fourth foliation explicitly. Using Serre's GAGA and Quillen-Suslin theorem, we show that for a suitable field extension K of k such a module over K[x,y] is actually generated by two elements, and therefore, such curves are free divisors in the sense of K. Saito. After performing Groebner basis for this module, we observe that in many well-known examples, K=k.