Limit Cycles as Topological Charges: A Lie Algebroid Cohomology Approach to Hilbert's 16th Problem

Abstract

This paper presents a framework that reformulates the counting problem of isolated compact closed leaves (limit cycles) in polynomial foliations as a topological charge pairing problem on a singular adapted Lie algebroid, providing an explicit upper bound for the finiteness of limit cycles in the second part of Hilbert’s 16th problem. Let X = (P, Q) be an n-th degree polynomial vector field on RP2 . We construct the log tangent bundle A = T f M(− log E) on the blow-up space f M = BlSing(X) RP2 , where E = S p∈Sing(X) Ep is the union of exceptional divisors. The vector field X is lifted to a global nondegenerate section sX of A. We define the closed 1-form ωtop = P dQ − Q dP P2 + Q2 , and prove that for each isolated limit cycle γ, the pairing R γ ωtop is a positive integer multiple of 2π, interpretable as a topological charge. By the generalized residue theorem on the blow up space, the total charge of all limit cycles is controlled by the first Chern class of A: X γ∈Liso Z γ ωtop ≤ 2π · ⟨c1(A), [Mf]⟩. This yields #Liso ≤ ⟨c1(A), [Mf]⟩. For n = 2, we compute the upper bound explicitly and verify compatibility with Shi Songling’s 4-limit-cycle example. A Lean 4 formal verification skeleton is provided. 本文提出一个框架,将多项式叶状结构中孤立紧致闭叶的计数问题重新表述为奇异适配 李代数胚上的拓扑荷配对问题,给出希尔伯特第 16 问题第二部分中极限环有限性的一个显式 上界估计。设 X = (P, Q) 为 RP2 上的 n 次多项式向量场。构造爆破空间 Mf = BlSing(X) RP2 上的对数切丛 A = T f M(− log E),其中 E = S p∈Sing(X) Ep 为所有例外除子之并。向量场 X 被提升为 A 的全局非退化截面 sX。定义 A 上的闭 1-形式 ωtop = P dQ − Q dP P2 + Q2 , 并证明:对每个孤立极限环 γ,配对值 R γ ωtop 为 2π 的正整数倍,即拓扑荷。由爆破空间上 的广义残数定理,所有极限环的总电荷被 A 的第一陈类控制: X γ∈Liso Z γ ωtop ≤ 2π · ⟨c1(A), [Mf]⟩. 由此得到 #Liso ≤ ⟨c1(A), [Mf]⟩。对 n = 2 的二次系统,显式计算 c1(A) 的上界并与史松龄 4 个极限环例子进行相容性分析。本文给出 Lean4 形式化验证骨架。