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Math-45 · mathematics · Algebra

Hilbert's Sixteenth Problem

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Statement

Definitions and mathematical statement

A planar polynomial vector field of degree at most $n$ is \[ \dot x=P(x,y),\qquad\dot y=Q(x,y),\qquad P,Q\in{\mathbb{R}}[x,y],\quad\max(\deg P,\deg Q)\le n. \] A limit cycle is an isolated periodic orbit, not an arbitrary member of a continuous family of periodic trajectories. Let $H(n)$ be the supremum of the number of limit cycles over this class. The problem asks for finiteness of $H(n)$, its maximal value, and the possible relative configurations of these cycles for each degree. A bound depending on the individual coefficients is not a uniform degree bound.

Formulation and concept references: [S1].

Short English statement

How many isolated periodic motions can a planar polynomial differential equation have, and how can those cycles be arranged?

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Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Math-45-A2 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-45

The quintic Liénard paper determines a maximum of two cycles for its specified subclass. The other selected manuscript asserts a finite bound for all planar polynomial vector fields at each fixed degree. Neither claim determines the sharp degree-by-degree maximum or the possible arrangements of cycles asked for here.

Hilbert's Sixteenth Problem

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.

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Math-45-A1 · No claim recorded · LLM-generated · gpt 6 astra pro ·

gpt 6 astra pro attempt on Math-45

Hilbert's Sixteenth Problem

Definitions and mathematical statement

A planar polynomial vector field of degree at most $n$ is \[ \dot x=P(x,y),\qquad\dot y=Q(x,y),\qquad P,Q\in{\mathbb{R}}[x,y],\quad\max(\deg P,\deg Q)\le n. \] A limit cycle is an isolated periodic orbit, not an arbitrary member of a continuous family of periodic trajectories. Let $H(n)$ be the supremum of the number of limit cycles over this class. The problem asks for finiteness of $H(n)$, its maximal value, and the possible relative configurations of these cycles for each degree. A bound depending on the individual coefficients is not a uniform degree bound.

Formulation and concept references: [S1].

Short English statement

How many isolated periodic motions can a planar polynomial differential equation have, and how can those cycles be arranged?

Research attempt

Current frontier.
Literature check: 27 September 2026. The distinction between finiteness for one vector field and a bound uniform in all its degree-$n$ coefficients is essential. The latter is not supplied by the former. The formulation reference studies local cyclicity through Melnikov functions and Lyapunov constants, rather than the entire global configuration problem; its journal record gives a degree-six local lower bound of 44. [S1], [E1]. The 2024 Buzzi--Novaes critique rules out the advertised quadratic formula for $H(n)$: the established lower-growth estimate is already of order $n^2\log n$. Thus a newspaper report about that formula is not evidence that the present target has been solved. [S2]. A 2026 manuscript investigates separable Chebyshev replication and its quadratic ceiling when no additional cycles are created. It is a useful construction-level comparison, not a general upper bound. [E2].
Attack route.
There are three distinct possibilities: construct increasingly many cycles at a fixed degree, obtain a uniform zero bound for suitable return maps, or cover the compactified coefficient space by neighborhoods with uniformly bounded cyclicity. The first must keep the degree fixed; the second must handle degenerate return maps; and the third needs local bounds at every limiting vector field, including those with centers, singular curves, and behavior at infinity. We pursue a concrete angular-return calculation, then test whether its uniformity survives degeneration. The computations below are deductions for this notebook; elementary ingredients and the restricted conclusions are not claimed to be historically new.
Attempt: an exactly solvable diagnostic family.
Set $s=x^2+y^2$ and consider \[ P=xF(s)-yG(x,y),\qquad Q=yF(s)+xG(x,y), \tag{40.1} \] where $F$ is a real polynomial and $G$ is polynomial and strictly positive on the closed annulus under consideration. In polar coordinates, \[ \dot r=rF(r^2),\qquad \dot\theta=G(r\cos\theta,r\sin\theta). \tag{40.2} \] Consequently every positive root $a$ of $F$ whose circle lies in the annulus gives a periodic orbit $r=\sqrt a$. These are the only periodic orbits there: away from a root, $r$ is strictly monotone, and a trajectory cannot cross an equilibrium of the scalar equation for $r$ by uniqueness. If $F\not\equiv0$, each such circle is isolated, even for a multiple root. If $F\equiv0$, the annulus is filled by periodic orbits and none is a limit cycle. Thus this family has exactly as many limit cycles in the annulus as $F$ has distinct positive roots there.

For a simple root $a$, let \[ T_a=\int_0^{2\pi}\frac{d\theta} {G(\sqrt a\cos\theta,\sqrt a\sin\theta)}. \] The transverse Floquet multiplier is \[ \mu_a=\exp\left(2aF'(a)T_a\right). \tag{40.3} \] Indeed the linearization of $rF(r^2)$ at $\sqrt a$ is $2aF'(a)$, constant along the orbit. Equivalently, differentiating $dr/d\theta=rF(r^2)/G$ at the root gives $2aF'(a)/G$, because the term differentiating $1/G$ is multiplied by $F(a)=0$. This also checks that the nonconstant angular speed has not been silently ignored.

Take $G=1$ and $F(s)=c\prod_{j=1}^{m}(s-a_j)$, with $0<a_1<\cdots<a_m$ and $c\ne0$. This is a degree-$2m+1$ field with precisely $m$ nested hyperbolic limit cycles, and their stability alternates. The example checks existence, isolation, degree, and relative position explicitly. It is only a lower bound for the unrestricted class; in particular it is nowhere near the known large lower bounds cited above.

Attempt: first-order angular return maps.
Now perturb the linear center by arbitrary degree-at-most-$n$ polynomials: \[ \dot x=-y+\varepsilon p(x,y),\qquad \dot y=x+\varepsilon q(x,y). \] On a fixed compact annulus $0<r_-\le r\le r_+<\infty$, write \[ R(r,\theta)=p(r\cos\theta,r\sin\theta)\cos\theta+ q(r\cos\theta,r\sin\theta)\sin\theta, \] \[ T(r,\theta)=\frac{q(r\cos\theta,r\sin\theta)\cos\theta- p(r\cos\theta,r\sin\theta)\sin\theta}{r}. \] For sufficiently small $|\varepsilon|$, the angular velocity $1+\varepsilon T$ stays positive and \[ \frac{dr}{d\theta}=\frac{\varepsilon R}{1+\varepsilon T}. \] Smooth dependence of solutions of this scalar ODE, uniformly on a slightly larger compact annulus for $0\le\theta\le2\pi$, gives the $C^1$ expansion \[ \Pi_\varepsilon(r)-r=\varepsilon M(r)+O(\varepsilon^2), \qquad M(r)=\int_0^{2\pi}R(r,\theta)\,d\theta. \tag{40.4} \] For completeness, the zeroth-order trajectory is constant; replacing the perturbed trajectory in $R$ by this constant changes its value by $O(\varepsilon)$ by Gronwall's inequality. Multiplying by the exterior $\varepsilon$ gives the stated remainder. Differentiating the scalar ODE in the initial radius proves the same bound for the first derivative. The constants depend on the annulus and bounded coefficients of $p,q$.

Parity now supplies an exact algebraic simplification. A monomial $x^iy^j$ in $p$ contributes $r^{i+j}\int\cos^{i+1}\theta\sin^j\theta\,d\theta$, which vanishes unless $i$ is odd and $j$ even. The analogous term from $q$ survives only when $i$ is even and $j$ odd. Hence \[ M(r)=rA(r^2),\qquad \deg A\le \left\lfloor\frac{n-1}{2}\right\rfloor. \tag{40.5} \] If all positive roots of $A$ in the annulus are simple and none is on its boundary, the implicit function theorem gives exactly one nearby limit cycle for every such root and no other cycles confined to the annulus, for sufficiently small nonzero $\varepsilon$. To justify "no other", remove small disjoint neighborhoods of the roots. On the remaining compact set $|M|$ has a positive minimum, and $40.4$ excludes return-map zeros. Inside each root neighborhood, $M'$ is bounded away from zero, so the $C^1$ remainder gives uniqueness as well as existence. An orbit making several full turns would give a periodic point of the strictly increasing scalar return map. Such a point must be fixed: if $\Pi(r)>r$ or $\Pi(r)<r$, successive defined iterates remain strictly ordered and cannot return to $r$. Thus multiple-turn periodic points do not evade the zero count.

As a check, choose $p=x(s-1)(s-4)$ and $q=y(s-1)(s-4)$. Then $M(r)=2\pi r(r^2-1)(r^2-4)$, and the two cycles at radii 1 and 2 are in fact exact for every nonzero $\varepsilon$. Their transverse exponents have opposite signs. The accompanying symbolic script verifies the polar identities and these signs exactly.

Stress test: why this does not produce a degree bound.
First, $40.5$ bounds zeros of the first nonzero displacement term only when that term is actually the first-order one. Taking $p=-yB(s)$ and $q=xB(s)$ gives $R=0$ and $M\equiv0$: the perturbation changes only angular speed, leaving a center. More generally $M\equiv0$ does not imply that the full return map is the identity, and higher-order terms can control the bifurcation. Bounding their order of appearance and their zeros uniformly is not supplied by the calculation. This is precisely why the local-cyclicity literature uses Lyapunov constants and higher-order information. [S1], [E1].

Second, the annulus estimates are not uniform as $r_-\downarrow0$, $r_+\uparrow\infty$, or the angular velocity approaches zero. Cycles may approach a singular point, a separatrix configuration, or infinity instead of one of the regular return-map charts. A fixed compact annulus is not a compactification of all these phenomena.

Here is an exact compactness reduction which isolates the omitted step. Normalize a nonzero degree-at-most-$n$ coefficient vector to Euclidean norm one; multiplying a vector field by a positive constant preserves all its orbits. The normalized vectors form a compact sphere $K_n$. Suppose that for every $v\in K_n$ there were an open neighborhood $U_v$ and an integer $B_v$ such that every vector field with coefficient vector in $U_v$ had at most $B_v$ limit cycles in the entire plane. A finite subcover gives \[ H(n)\le\max_{1\le j\le N}B_{v_j}<\infty. \tag{40.6} \] This implication is proved, but its hypothesis is unproved at degenerate fields. Pointwise finiteness does not imply this local boundedness: the function on $\{0\}\cup\{1/j:j\ge1\}$ given by $b(0)=0$ and $b(1/j)=j$ is finite at every point of a compact space and has no finite bound. No upper semicontinuity of the cycle count has been established here.

Third, even $40.6$ would address only finiteness, not the target's request for the exact maximum and all realizable configurations. Nested circles in $40.1$ do not classify nonnested or mixed configurations. A replication scheme whose degree increases at every step cannot disprove finiteness at one fixed degree; compare the explicitly degree-increasing 2026 construction. [E2].

Outcome.
Outcome: Exploratory approach with unresolved gap. We obtained an exact cycle count and Floquet formula in an angular-radial subclass, a uniform first-order return-map calculation on regular compact annuli, and a precise compactness implication for global finiteness. The stress tests identify loss of control when the first Melnikov function vanishes or return charts degenerate. There is no new global bound on $H(n)$, no fixed-degree unbounded construction, and no classification of all configurations. A resolution would need uniform control at every limiting field and, beyond finiteness, a separate sharpness and realization analysis. Historical novelty of the diagnostic lemmas is not asserted.

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