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Math-2190 · mathematics · Geometry

Hilbert's 16th Problem: Topology of Algebraic Curves and Limit Cycles

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For a planar polynomial differential equation $\dot x=P(x,y)$, $\dot y=Q(x,y)$ with $\deg P,\deg Q\leq n$, determine the largest possible number of isolated periodic orbits, called limit cycles, and their possible relative positions. The other part of the problem asks for the topology and arrangements of connected components of real algebraic curves and surfaces of prescribed degree. The degree bound is fixed while the coefficients vary.

Formulation references: [S1], [S2].

Short English statement

Determine the topology of real algebraic curves and the maximum number of polynomial-flow limit cycles at each fixed degree.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2190-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2190

Manuscript 1 (family 143, 2026-09-24): The exact two-cycle maximum is determined for classical quintic Lienard systems only. Arbitrary degree-five planar vector fields, all other degrees and arrangements are outside the theorem.

Manuscript 2 (family 143, 2026-09-24): Uniform finiteness for each degree answers the finiteness clause of the second Hilbert-sixteenth problem, but not the sharp maximum or relative arrangements; record 46 also bundles algebraic-curve topology.

Hilbert's 16th Problem: Topology of Algebraic Curves and Limit Cycles

OpenAI's claim

Manuscript 1 (family 143, 2026-09-24): A classical Liénard system with arbitrary real primitive polynomial of degree at most five is claimed to have at most two isolated periodic orbits, and two are claimed to occur. Manuscript 2 (family 143, 2026-09-24): 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.

Mathematical statement of the claimed result

Mathematical claim 1.
For any real polynomial $F$ of degree at most five, consider \[ \dot x=y-F(x),\qquad\dot y=-x. \] OpenAI claims at most two limit cycles in $\mathbb R^2$, with examples attaining two. A cycle means the geometric image of a nonconstant periodic trajectory isolated among periodic orbits; multiplicity and stability do not affect the count. Periodic trajectories of a center are excluded.
Mathematical claim 2.
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): A quadratic midpoint model tracks fitted slope and curvature by transport equations. Endpoint variation, Riccati linearization, and a Schwarzian identity control the width dependence of the model coefficients, allowing the two-cycle argument without the odd-polynomial restriction of the older theorem.

Manuscript 2 (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. The documented statement covers the main claim of manuscript 1. Its selected declaration also covers the clause used for this problem. Manuscript 2 is not among the papers selected by that page, so the family link does not formalize its main conclusion. It does not establish formal coverage of the complete problem clause. Comparator files: QuinticLienard.lean. Declarations: OAI.QuinticLienard.main.

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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