Hilbert's sixteenth problem: uniform bounds for limit cycles

Claims a finite cap, set by degree alone, on limit cycles of planar polynomial systems.

2 papers September 24, 2026 Explained by AI

A limit cycle is a closed path of a moving point with no other closed path right next to it. The first paper claims that polynomial rules of one degree share a finite cap on these loops, with no formula. The second paper studies Liénard systems, a simple family of such rules built from one polynomial F. It proves at most two limit cycles when F has degree at most 5, and shows two can occur. Lean, a program that checks each step of a proof, has checked this second result.

First step

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Significance

Landmark

If the long paper's argument holds, it would establish uniform boundedness in the second part of Hilbert's 16th problem. Only the quintic Liénard result has a Lean check.

Trust

Formal proof listed

Field

Dynamical systems and ergodic theory

2 papers

The explanation

Picture a point moving in a flat plane. At each spot, a rule says which way to go and how fast. The second of the two papers builds one such rule. It uses a small push size ε and the polynomial f(x) = 4x − (20/3)x³ + (8/5)x⁵. With no push, every path is a perfect circle around the centre. With a small push, a path gains or loses a little energy on each lap.

The paper computes that gain for a lap of radius s. For a small push it is about ε times Q(s). Here Q(s) = −π s²(4 − 5s² + s⁴)Source 1. The bracket factors as (s² − 1)(s² − 4). So Q is zero exactly at radius 1 and radius 2. At radius 0.5, Q is about −2.21, so small paths lose energy and shrink. At radius 1.5, Q is about +15.46, so those paths grow. At radius 2.5, Q is about −231.9, so wide paths shrink back. Paths are squeezed toward a loop near radius 2 and pushed away from a loop near radius 1.

Those two loops are limit cycles. A limit cycle is a closed path that the point repeats forever. No other closed path lies right next to it. In many cases, nearby paths spiral toward it or away from it. The paper shows that this example, for small positive ε, has two such loops. The inner one repels and the outer one attractsSource 2.

How many limit cycles can a rule have? David Hilbert asked this in his famous list of problems, for rules built from polynomials. The degree of the rule is the highest power of x or y that appears. Two questions hide inside his problem. One asks whether each single rule has only finitely many limit cycles. The other asks for one ceiling that works for every rule of a given degree. Dulac claimed the first in 1923. A gap in his argument was later found. Écalle and Ilyashenko gave new proofs in 1990 to 1992. In 2025, Yeung flagged a problem in one step of Ilyashenko's 1991 proof, though not a counterexample. Even then, finiteness for each rule does not bound the counts as the coefficients changeSource 3.

That is the obstacle. As the numbers in a rule change, limit cycles can appear out of degenerate configurations. They can also drift arbitrarily far from the centreSource 4. Earlier work, such as Ilyashenko and Yakovenko's and Kaloshin's, counted cycles near such shapes only under extra conditions. The first paper claims to remove those conditions. It claims that for every degree d there is a finite number B(d). No polynomial rule of degree at most d has more than B(d) limit cycles anywhere in the planeSource 5. The paper does not give a formula for B(d)Source 6. It shows a ceiling exists, but not how high it sits.

The second paper looks at one family where the exact ceiling can be pinned down. These are classical Liénard systems: x′ = y − F(x) and y′ = −x, with F a polynomial. In 1977, Lins, de Melo and Pugh proposed at most ⌊(n − 1)/2⌋ limit cycles when F has degree n. For degree 5 that guess gives 2. Rychkov proved 2 in 1975 when F has only odd powers. Counterexamples later broke the guess in degree 7 and in degree 6. Li and Llibre proved the degree-4 bound of 1. Degree 5 with all powers allowed stayed open. An August 2026 preprint by Chen, Li, Zhang and Zhang still reported it as open.

The second paper proves that when F has degree at most 5, the system has at most two limit cyclesSource 7. It puts no sign or symmetry condition on FSource 8. The example above shows two can occur, so two is the exact maximumSource 7. In that example, the main theorem rules out a third loop. Because F may have even powers, the problem cannot be reduced to Rychkov's odd caseSource 9.

How sure is this? A computer proof checker called Lean has checked the Liénard theorem for F of degree at most 5. That check covers both the ceiling of two and an example with exactly two. The general bound for every degree has no Lean check. That claim rests on the paper alone. No specialist review of either paper has been published.

What it doesn't show. This does not give a formula or a value for the general cap B(d) in any degree.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Landmark only if the long paper's argument holds, and no one outside the release has checked it. Lean covers only the narrow degree-5 Liénard result. Neither paper gives a value for the general cap.

Words used

Vector field: A rule that gives a direction and speed at every point of the plane.
Limit cycle: A closed path repeated forever, with no other closed path right next to it.
Degree: The highest power of the variables appearing in a polynomial rule.
Liénard system: The rule x′ = y − F(x), y′ = −x, with F a polynomial. It is equivalent to x″ + F′(x)x′ + x = 0.
Uniform bound: One cap that works for every rule of a given degree at once.
Lean: A computer program that checks each step of a formal proof.

The original papers (2)

Uniform bounds for planar polynomial limit cycles (PDF, opens in a new tab)
September 24, 2026
Two limit cycles for quintic Liénard systems (PDF, opens in a new tab)
September 24, 2026 Formal proof listed

Lean notes

All quoted lines (9)

f(x)=4x-\frac203x^3+\frac85x^5. … Q(s,0)=-\pi s^2(4-5s^2+s^4).
Two limit cycles for quintic Liénard systems Back to text
Thus both cycles are hyperbolic: the inner one repels and the outer one attracts.
Two limit cycles for quintic Liénard systems Back to text
Even finiteness for every member of a family does not by itself bound the numbers occurring as its coefficients vary.
Uniform bounds for planar polynomial limit cycles Back to text
The difficulty is to bound these zeros while the coefficients vary and the passages making up the return map degenerate.
Uniform bounds for planar polynomial limit cycles Back to text
For each integer $d\geq1$ , there is a finite nonnegative integer $B(d)$ such that every real planar polynomial vector field of degree at most $d$ has at most $B(d)$ limit cycles in the whole plane.
Uniform bounds for planar polynomial limit cycles Back to text
It does not furnish an effective formula for $B(d)$ .
Uniform bounds for planar polynomial limit cycles Back to text
Thus its exact maximum is two.
Two limit cycles for quintic Liénard systems Back to text
No parity or coefficient-sign assumption is imposed on $F$ .
Two limit cycles for quintic Liénard systems Back to text
The even coefficients prevent reduction to the odd quintic family, and a small-amplitude calculation cannot by itself bound distant cycles.
Two limit cycles for quintic Liénard systems Back to text