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Math-2693 · mathematics · Dynamical systems

Uniform finiteness of planar polynomial limit cycles

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every integer $d>=1$, does there exist a finite B(d) such that every real planar polynomial vector field $x-prime=P(x,y)$, $y-prime=Q(x,y)$, with max(deg P,deg $Q)<=d$, has at most B(d) geometrically distinct isolated periodic orbits throughout the plane?

Short English statement

For every integer $d>=1$, does there exist a finite B(d) such that every real planar polynomial vector field $x-prime=P(x,y)$, $y-prime=Q(x,y)$, with max(deg P,deg $Q)<=d$, has at most B(d) geometrically distinct isolated periodic orbits throughout the plane?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2693-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2693

Manuscript 1 (family 143, 2026-09-24): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Uniform finiteness of planar polynomial limit cycles

OpenAI's claim

For each polynomial degree, a finite bound on the number of isolated periodic orbits of all real planar polynomial vector fields of that degree is claimed. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For each integer $d\ge1$, OpenAI asserts existence of a finite nonnegative integer $B(d)$ satisfying \[ \#\{\text{isolated periodic-orbit images of }\dot x=P(x,y),\ \dot y=Q(x,y)\}\le B(d) \] for every pair of real polynomials $P,Q$ of total degree at most $d$. The count covers the whole plane and ignores multiplicity. The theorem is a qualitative uniform bound; it does not specify a sharp numerical value of $B(d)$.

Outline of the argument

Manuscript 1 (family 143, 2026-09-24): Nested complex domains separate the required asymptotic scales, while high real jets retain a fixed coherence order through rapidly rotating dependent scales. Proper barriers and generic tilts reduce parameters in a derivative-stable analytic class; finite-dimensional counting then turns absolute finiteness into a uniform degree bound.

Formal verification

For family 143, the scope page documents the following selections. The selection concerns degree-at-most-five Liénard systems: two isolated cycles are the maximum and that maximum occurs. It is not the arbitrary-degree planar polynomial bound. Manuscript 1 is not among the papers selected by that page, so the family link does not formalize its main conclusion.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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