On the holomorphic foliations admitting a common invariant algebraic set

2025/05/16 by Guangfeng Dong, Dong, Guangfeng, Cheng-Ze Lu +1
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Polynomial and algebraic computation #Primary: 37F75 #Secondary: 32M25

paper · pdf · doi:10.48550/arxiv.2505.11172

openalex publication_date 2025/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the holomorphic foliations admitting a common invariant algebraic set C defined by a polynomial f in \mathbbK[x₁,x₂,...,x_n] over any characteristic 0 subfield \mathbbK⊆ℂ. For the \mathbbK[x₁,x₂,...,x_n]-module V_f of vector fields generating foliations that admit C as an invariant set, we provide several conditions under which the module V_f can be freely generated by a minimal generating set. In particular, when n=2 and f is a weakly tame polynomial, we show that the \mathbbK[x,y]-module V_f is freely generated by two polynomial vector fields, one of which is the Hamiltonian vector field induced by f, if and only if, f belongs to the Jacobian ideal ⟨ f_x, f_y⟩ in \mathbbK[x,y]. Our proof employs a purely elementary method.

Citations

• Characterization of the kukles polynomial differential systems having an invariant algebraic curve
• Number of limit cycles for planar systems with invariant algebraic curves
• Quadratic Planar Differential Systems with Algebraic Limit Cycles via Quadratic Plane Cremona Maps

Related

• Algebraic curves and foliations
• Polynomial vector fields with algebraic trajectories
• Invariant varieties for polynomial dynamical systems
• Holomorphic vector fields tangent to foliations in dimension three
• Vector fields and foliations associated to groups of projective automorphisms