Polynomial Bound for Limit Cycles of Planar Polynomial Vector Fields

Polynomial Bound for Limit Cycles of Planar Polynomial Vector Fields

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. It does not establish formal coverage of the complete problem clause.

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

• OpenAI, Uniform bounds for planar polynomial limit cycles, 2026-09-24; citation key OAI:uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:uniform.
• Family 143: Lean scope.
• OpenAI catalogue entry.

Polynomial Bound for Limit Cycles of Planar Polynomial Vector Fields

Definitions and mathematical statement

For real polynomials $P,Q$ of degree at most $d$, a limit cycle of $\dot x=P(x,y)$, $\dot y=Q(x,y)$ is an isolated periodic orbit. The target asks whether there is one exponent $q>0$ with \[ \#\{\text{limit cycles of }(P,Q)\}\le d^q \] for every degree $d$ and every such system, with the nontrivial degree range $d\ge2$. The exponent is independent of all coefficients and of $d$. A finite but faster-than-polynomial degree bound would not answer this stronger quantitative question.

Formulation and concept references: [S1].

Short English statement

Can the number of limit cycles of every planar polynomial system be bounded by one universal power of its degree?

Research attempt

Current frontier.

Smale's Problem 13 asks for the universal polynomial degree bound in the catalogue, not merely finiteness for a fixed vector field. [S1], Problem 13. Buzzi--Novaes' 2024 analysis documents why a proposed quadratic formula for the Hilbert numbers conflicts with established lower constructions of order $d^2\log d$. That refutation does not refute a bound $d^q$ with some larger universal exponent. [E1], Sections 1--3. The present attempt actively constructs systems with many isolated cycles, keeps track of their degree, and checks whether the construction can outgrow every polynomial.

Attack routes.

A counterexample could come from many separated oscillating regions, from many nested cycles, or from a recursive construction whose cycle count grows faster than the cost in degree. A proof would need uniform control of zeros of return maps as coefficients vary and degenerate. Counting zeros of a polynomial surrogate is insufficient unless an exact relation with isolated periodic orbits is established. We combine the first two construction mechanisms explicitly, then calculate why that combination alone fails to give superpolynomial growth.

Attempt I: many wells with controlled algebraic degree.

Let $r\geq1$, and choose a real polynomial $f$ of degree $r$ with $r$ distinct real roots, for example $f(t)=\prod_{i=1}^r(t-i)$. Put \[ H(x,y)=f(x)^2+f(y)^2. \] There is $a>0$ such that $f^{-1}((-a,a))$ consists of $r$ disjoint intervals, each containing one root, and $f$ maps each interval smoothly and bijectively onto $(-a,a)$. To justify this globally, choose disjoint inverse-function neighborhoods of the simple roots. The proper function $|f|$ has a positive minimum on their complement after slightly shrinking their images; choose $a$ below that minimum. Thus no additional small-value component is omitted.

On each of the $r^2$ resulting rectangles, the map $(x,y)\mapsto(f(x),f(y))$ is a diffeomorphism onto $(-a,a)^2$. For every $0<c<a^2$, the level $H=c$ is therefore exactly $r^2$ disjoint smooth closed ovals, one in each rectangle. There are no other components, since $H<c$ forces both $|f(x)|,|f(y)|<a$. Its gradient is nonzero on every such level: in these local coordinates the level is a nonzero circle and the coordinate change is invertible.

Choose distinct values $0<c_1<\cdots<c_m<a^2$ and write $h(s)=\prod_{j=1}^m(s-c_j)$. For the quarter-turn matrix $J=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ define the polynomial vector field \begin{equation}\label{wsref-0-33-30-30-30-36-37-34-30-3a-66-69-65-6c-64} V=J\nabla H-h(H)\nabla H. \end{equation} Its degree is exactly \begin{equation}\label{wsref-0-33-30-30-30-36-37-34-30-3a-64-65-67-72-65-65} d=2r(m+1)-1. \end{equation} The highest-degree part comes from $h(H)\nabla H$; it is nonzero and has strictly greater degree than $J\nabla H$. Most importantly, \begin{equation}\label{wsref-0-33-30-30-30-36-37-34-30-3a-65-6e-65-72-67-79} \dot H=\nabla H\cdot V=-h(H)|\nabla H|^2. \end{equation} Every oval in $H=c_j$ is invariant. On it the vector field equals $J\nabla H$, which is everywhere nonzero. A smooth nonvanishing vector field on a compact circle traverses it in finite period, so these are periodic orbits, not merely invariant sets.

Attempt II: prove isolation and count all cycles.

Use the transverse coordinate $s=H-c_j$ near one oval. Linearizing (30006740:energy) along its period-$T_j$ trajectory $\gamma$ gives \[ \dot s=-h'(c_j)|\nabla H(\gamma(t))|^2s. \] Its return multiplier is \begin{equation}\label{wsref-0-33-30-30-30-36-37-34-30-3a-6d-75-6c-74-69-70-6c-69-65-72} \exp\!\left(-h'(c_j)\int_0^{T_j} |\nabla H(\gamma(t))|^2dt\right)\ne1. \end{equation} The integral is strictly positive and $h'(c_j)\ne0$. Thus every constructed periodic orbit is hyperbolic and isolated, and is genuinely a limit cycle in the definition of the problem.

There are no further nonconstant periodic orbits. A nonstationary trajectory cannot hit a critical point of $H$, because there $V=0$ and uniqueness of the smooth ordinary differential equation would make it stationary. Between consecutive roots of $h$, (30006740:energy) then has a fixed strict sign. A trajectory cannot cross a level $H=c_j$ either: that regular level is invariant and uniqueness prevents crossing. Consequently a periodic trajectory must have constant $H$ equal to one of the $c_j$, and every such level has already been counted. We have proved an exact count \begin{equation}\label{wsref-0-33-30-30-30-36-37-34-30-3a-63-6f-75-6e-74} K=mr^2 \end{equation} for this family. This conclusion is analytic, not inferred from a phase portrait or a numerical orbit integrator.

For a concrete example take $f(t)=t(t-1)$ and $c_1=1/100$. The map has four wells, and the above construction gives four hyperbolic limit cycles for a degree-seven field. The threshold can be checked explicitly: the only nonzero critical value of $f$ in magnitude is $1/4$, so one may take any $a<1/4$ with $c_1<a^2$. The companion script expands this example and verifies (30006740:energy) and the degree exactly.

Attempt III: can nesting beat the degree cost?

Combining (30006740:degree) and (30006740:count) gives \[ K=\frac{m}{4(m+1)^2}(d+1)^2 \leq\frac{(d+1)^2}{16}, \] since $(m+1)^2\geq4m$. Increasing the number of nested levels at each well consumes degree too quickly; this entire construction remains quadratic in degree. The calculation is a barrier for this chosen ansatz, not a universal upper bound. In particular, it cannot compete with the known $d^2\log d$ lower constructions recorded in [E1].

One may try to insert oscillatory factors with many real roots but low polynomial degree. For a genuine polynomial $h$, its number of distinct roots is at most its degree, which is exactly the cost in (30006740:degree). Replacing it by an oscillatory transcendental function would leave the problem's polynomial class. Alternatively, using many algebraic wells while pretending that their defining polynomial has fixed degree contradicts the same degree bookkeeping. These explicit failures explain why this counterexample search stops without suggesting that every possible construction is similarly bounded.

On the proof side, the exact identity (30006740:energy) is not available for a general polynomial vector field. Its Poincaré return map is analytic on its domain, but need not be a polynomial of degree controlled by $d$. Analytic functions can have arbitrarily many isolated zeros as their coefficients vary. To apply a zero-counting theorem one would need a uniform complexity bound for the actual return maps, including near separatrices and singular limiting systems. No such general bound is established here.

Stress test.

The construction counts isolated nonconstant periodic orbits, not centers, equilibria, or arbitrary compact algebraic components. Its multiplier verifies robustness under sufficiently small perturbations of a fixed constructed field, but the allowable perturbation size may depend on $r,m$ and the selected levels. That local robustness does not produce a uniform theorem over all coefficient space.

The catalogue permits any universal exponent $q>0$ and no coefficient- height dependence. A lower bound exceeding every quadratic polynomial rules out $q\leq2$, not all larger $q$. Conversely, per-field finiteness and normalization of the coefficient vector do not by themselves give a uniform bound: the normalized parameter space contains degenerate limits, and the cycle-counting function need not be bounded by a naive compactness argument. The critique in [E1] illustrates why changing the definition of a cycle to a computable surrogate would invalidate the target rather than solve it.

Outcome.

Outcome: Exploratory approach with unresolved gap.

A two-parameter polynomial family with exactly $mr^2$ hyperbolic limit cycles and degree $2r(m+1)-1$ has been constructed and proved to have no other periodic cycles. Its degree accounting shows that separated wells plus nested algebraic levels cannot yield a superpolynomial counterexample by this mechanism. The general return-map bound or a more efficient recursive counterexample construction remains missing. This family is not claimed as a new lower-bound record or a resolution.

Sources

• S1 Steve Smale, Mathematical Problems for the Next Century, 1998. https://www.fim.uni-passau.de/fileadmin/dokumente/fakultaeten/fim/lehrstuhl/muller/SmaleProblems1998.pdf. Catalogue formulation source.
• E1 Claudio A. Buzzi and Douglas D. Novaes. A note on a recent attempt to solve the second part of Hilbert's 16th Problem, arXiv:2411.09594, 14 November 2024. Discussion of the claimed quadratic formula and the established superquadratic lower constructions. https://arxiv.org/html/2411.09594. Consulted 27 September 2026 for the indicated result or scope.