What does this research mean for the field?

The number of isolated limit cycles in polynomial foliations can be explicitly bounded by reformulating the counting problem as a topological charge pairing on a singular adapted Lie algebroid, where the total charge is controlled by its first Chern class. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Establish an explicit finiteness upper bound on isolated limit cycles in polynomial foliations by reformulating the counting problem as a topological charge pairing on an adapted Lie algebroid.

September 8, 2026Open Access

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

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Establish an explicit finiteness upper bound on isolated limit cycles in polynomial foliations by reformulating the counting problem as a topological charge pairing on an adapted Lie algebroid.
Constructed a log tangent bundle over the blow-up space of the real projective plane at the singular locus of an n-th degree polynomial vector field.
Defined a differential 1-form yielding positive integer multiples of 2π across isolated limit cycles and applied a generalized residue theorem.
Formulated a Lean 4 formal verification skeleton to encode and check the theoretical framework.
Proved that the total number of isolated limit cycles is bounded by the evaluation of the first Chern class on the fundamental class of the blow-up space.
Explicitly computed the upper bound for quadratic vector fields (n = 2) and confirmed consistency with Shi Songling's four-limit-cycle construction.

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Implication

Theoretical study demonstrates an explicit upper bound for limit cycles in polynomial vector fields on the real projective plane, highlighting a topological resolution to Hilbert's sixteenth problem.

Cite This Study

Changmin Wei (2026) studied this question.

synapsesocial.com/papers/6a9fd7ab58e84d0ff5b46919 https://doi.org/10.5281/zenodo.22646353

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Implication

Theoretical study demonstrates an explicit upper bound for limit cycles in polynomial vector fields on the real projective plane, highlighting a topological resolution to Hilbert's sixteenth problem.

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

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Discussion

Implication

Cite This Study

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

Key Points

Discussion

Implication

Cite This Study

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