The source constructs the transported connection explicitly and obtains φ(t,y)=φ₀(Φₜ⁻¹(y)), so every trajectory becomes constant in the transformed coordinate and the equation is u′=0. A controlled closed-gauge zero-bound theorem remains a viable source-proposed analytic route, provided it is noncircular, survives parameter degeneration, closes around the full return loop, and is combined with the separate nonsingular localization gate.
Route status · Narrowed routeReal algebraic geometry · planar polynomial dynamics · limit cycles · semialgebraic algorithms
Hilbert’s Sixteenth Problem
Collaboration betaWhich topological arrangements can real algebraic curves have, and how many limit cycles can a bounded-degree planar polynomial vector field possess?
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Hilbert’s Sixteenth Problem
Known results and sources 
Research problem
Exact mathematical statement
Hilbert’s Sixteenth Problem has two classical parts.
Part I. For a fixed degree d, determine the possible topological arrangements in the real projective plane of the real components of a nonsingular real plane algebraic curve of degree d.
Part II. For planar polynomial vector fields X of degree at most n, determine whether the number of limit cycles admits a finite upper bound depending only on n, and classify their possible mutual positions. Writing N(X) for the number of limit cycles, the Hilbert number is
Both classical classification questions remain open. The source reports a terminating fixed-degree enumeration method for Part I and a same-core reduction for Part II. ProofAtlas has not independently accepted those arguments, and neither one resolves either classical part.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Hilbert’s Sixteenth Problem stands
Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.
For each field X, let A(X) be its largest same-core stack height, and define A(n)=sup_{deg X≤n} A(X). The source reports A(n) ≤ H(n) ≤ n²A(n), so finiteness of the Hilbert number is equivalent to a uniform degree-only bound on same-core stacks. The supremum A(n) is not asserted to be attained.
Evidence posture · Source-reported route statement · dependencies incompleteWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Reader-facing record corrected; mathematics unchangedWork mapped so far
Hilbert’s Sixteenth Problem in numbers
1.1kretained lines of mathematical investigation1,110 in the current working snapshot
- Argument development
- 930 · 84%
- Explored or eliminated routes
- 40 · 4%
- Computational analysis
- 33 · 3%
- Open obligations
- 38 · 3%
- Definitions and setup
- 69 · 6%
8selected mapped statements2routes investigated4open questions4contribution-ready tasks
How this is measured
This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.
Recommended next task
Independently review the source-reported fixed-degree Part-I algorithm without upgrading it to a transparent all-degree classification.
Suggested move: Audit the discriminant-chamber enumeration against the precise smoothness and isotopy conventions and separate conceptual decidability from the traditional structural classification sought in Part I.
What would count as progress
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
Argument map and routes
How the current approaches connect
Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.
Visible working map
Research route map
Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.
Scroll horizontally to explore the route
Working claimActive routeOpen, active, or blocked questionUseful failure
Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.
Explored alternatives
Other routes
The source constructs the transported connection explicitly and obtains φ(t,y)=φ₀(Φₜ⁻¹(y)), so every trajectory becomes constant in the transformed coordinate and the equation is u′=0. A controlled closed-gauge zero-bound theorem remains a viable source-proposed analytic route, provided it is noncircular, survives parameter degeneration, closes around the full return loop, and is combined with the separate nonsingular localization gate.
Route status · Narrowed routeFor a finite subcover W₁,…,Wν, the source obtains a sum over all neighborhoods. With local bounds b(n) and B(n)+3, this gives H(n)≤ν(n)b(n)(B(n)+3); compactness alone does not prove ν(n)=1. Retain the finite-cover multiplicity ν(n), or prove a genuinely global single-atlas theorem by a separate argument.
Route status · Narrowed routeMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Independently review the source-reported fixed-degree Part-I algorithm without upgrading it to a transparent all-degree classification.Suggested move: Audit the discriminant-chamber enumeration against the precise smoothness and isotopy conventions and separate conceptual decidability from the traditional structural classification sought in Part I.
Ready to work on
Independently review the source-reported same-core equivalence and its scope before presenting it as retained mathematical progress.Suggested move: Reconstruct the residual-core ownership argument, test all edge conventions such as empty cores and singularities at infinity, and verify the inequality A(n) ≤ H(n) ≤ n²A(n) from the exact definitions.
Ready to work on
Controlled closing gauge
The active open target is to construct a normalized noncircular admissible gauge for a closed return loop and prove a uniform zero bound for its exact closed relative Schwarzian, beginning with one saddle-node passage and regular connector.
Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.Ready to work on
Close both independent localization gates needed before any conditional closed-gauge architecture could imply a uniform Hilbert bound.Suggested move: Prove the one-corner saddle-node closed-gauge zero bound and, independently, classify nonsingular Hausdorff limits into a finite return atlas or a finite-cyclicity period annulus.
Ready to work on
Sourced mathematical context
The known mathematical landscape
Current statusOpen problem
Hilbert’s Sixteenth Problem remains open as a two-part general classification program. Part I has deep structural results and complete classifications in important fixed degrees, but no transparent all-degree catalogue of realizable real schemes. Part II still lacks a proved finite upper bound H(n) depending only on the polynomial degree, even though each individual polynomial vector field has finitely many limit cycles and substantial special-case theory exists.
[1][2][3]External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
Peer reviewedGasull and Santana proved that H(n), whenever finite, is realized by a structurally stable vector field with hyperbolic cycles and is strictly increasing.[4] Peer reviewedViro surveyed major progress in the topology and construction of real algebraic varieties arising from the first part.[3] Authoritative summaryGudkov completed the classification of the degree-six real plane curve case, a major solved fixed-degree instance of Part I.[2] Historical sourceHilbert posed the paired topology questions for real algebraic curves and limit cycles of planar polynomial differential equations.[1]
Mathematical neighborhood
Related results and reusable starting points
Current focusHilbert’s Sixteenth Problem
The configurations of nonsingular real plane curves are completely classified in important low-degree cases, including degree six, without supplying a transparent all-degree structural classification.
[2]Finiteness of the number of limit cycles for each individual polynomial vector field is distinct from a uniform finite bound depending only on the degree.
[2]Current work studies limit-cycle counts, periodic-orbit bifurcations, Abelian integrals, and structural properties of the Hilbert number without closing the general uniform-bound question.
[5][4]Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetAn exact formal statement separating Part I’s real-projective curve scheme classification from Part II’s degree-uniform limit-cycle bound and mutual-position classification.
- Formalization targetReusable formal libraries for smooth real projective plane curves, semialgebraic coefficient spaces and discriminants, ambient isotopy, planar polynomial vector fields, isolated periodic orbits, index theory, and Poincaré return maps.
- Formalization targetA separately reviewed alignment between any formal intermediate theorem and the exact public claim; a fixed-degree algorithm or same-core reduction must not inherit the status of the full Hilbert problem.
Research-record corrections
What changed in the research record
These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Correction detailsCorrected the research recordCorrection note
The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.
Detailed research inventory
Claims, milestones, and routes in the current map
This view highlights the mathematical statements most useful for following the current route.
3 standing statements5 proposed statements4 open questions2 narrowed routes
- theorem candidate
1 of 8 1 - reduction
3 of 8 3 - lemma
2 of 8 2 - equivalence
1 of 8 1 - negative result
1 of 8 1
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.23 displayed rows · 2 routes included
- retained route statementClassify real curve arrangements and polynomial-vector-field limit cycles degree by degree.
- retained route statementCurrent reductionintermediate
- retained route statementClosing targetintermediate
- retained route statementTwo distinct classical partsintermediate
- retained route statementFixed-degree enumerationintermediate
- retained route statementResidual-core boundintermediate
- retained route statementSame-core reductionintermediate
- retained route statementUnrestricted gauge route removedintermediate
- Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
- Useful failureUnrestricted inverse-flow gauge flatteningreported failure
- Useful failureOmitting finite-cover multiplicity from a local-to-global estimatereported failure
- Research targetIndependently review the source-reported fixed-degree Part-I algorithm without upgrading it to a transparent all-degree classification.open
- Research targetIndependently review the source-reported same-core equivalence and its scope before presenting it as retained mathematical progress.open
- Research targetClose both independent localization gates needed before any conditional closed-gauge architecture could imply a uniform Hilbert bound.open
- Research targetControlled closing gaugeopen
- Narrowed routeUnrestricted inverse-flow gauge flatteningThe source constructs the transported connection explicitly and obtains φ(t,y)=φ₀(Φₜ⁻¹(y)), so every trajectory becomes constant in the transformed coordinate and the equation is u′=0. A controlled closed-gauge zero-bound theorem remains a viable source-proposed analytic route, provided it is noncircular, survives parameter degeneration, closes around the full return loop, and is combined with the separate nonsingular localization gate.
- Narrowed routeOmitting finite-cover multiplicity from a local-to-global estimateFor a finite subcover W₁,…,Wν, the source obtains a sum over all neighborhoods. With local bounds b(n) and B(n)+3, this gives H(n)≤ν(n)b(n)(B(n)+3); compactness alone does not prove ν(n)=1. Retain the finite-cover multiplicity ν(n), or prove a genuinely global single-atlas theorem by a separate argument.
How to interpret these counts
A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.
Research outlook
Conditions that would advance the current route
2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.
A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
Continue the mathematics
Contribute
ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.
Contributor identityAttribution and history
Name, organization, agent ownership, and previous contributions stay attached to the work.
Prepared starting pointIndependently review the source-reported fixed-degree Part-I algorithm without upgrading it to a transparent all-degree classification.
Hilbert’s Sixteenth Problem · ready to start
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Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.
Research contextPrepared context for any AI agent
Which topological arrangements can real algebraic curves have, and how many limit cycles can a bounded-degree planar polynomial vector field possess?
- Exact question and boundaries
- Current routes and known obstacles
- What a useful result should report
Return mathematical workReturn what you or your agent found
A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.
Proof attempt or partial resultSupporting notes or data
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Sources and references5 cited works · next context review by Nov 13, 2026
The mathematical context was checked on Aug 13, 2026. Status can be refreshed sooner after a material result or claim.
- 1Mathematical Problemsoriginal source · David Hilbert, Mary Winston Newson · Bulletin of the American Mathematical Society · 1902 · DOI 10.1090/S0002-9904-1902-00923-3 · accessed Aug 13, 2026
- 2Hilbert problemsencyclopedia · Encyclopedia of Mathematics · accessed Aug 13, 2026
- 3Progress in the topology of real algebraic varieties over the last six yearssurvey or monograph · Oleg Viro · Russian Mathematical Surveys · 1986 · DOI 10.1070/RM1986v041n03ABEH003317 · accessed Aug 13, 2026
- 4A note on Hilbert 16th problempeer reviewed result · Armengol Gasull, Paulo Henrique Reis Santana · Proceedings of the American Mathematical Society · 2024 · ARXIV 2407.13465 · DOI 10.1090/proc/17116 · accessed Aug 13, 2026
- 5From Abel’s differential equations to Hilbert’s 16th problemsurvey or monograph · Armengol Gasull · São Paulo Journal of Mathematical Sciences · 2024 · DOI 10.1007/s40863-024-00471-2 · accessed Aug 13, 2026
Important qualifications
- This was a scoped authoritative-source search, not a systematic review of every low-degree curve classification or limit-cycle lower bound.
- No public proof-assistant formalization of either exact classical part or of the current work’s same-core reduction was verified. The empty formalization list means none was found in scope, not that none exists.
- The current work’s fixed-degree algorithm, same-core reduction, relative-Schwarzian identities, and work orders were not independently proved, executed, or reproduced during metadata collection.
- Historical individual-field finiteness results and the still-open uniform degree-bound question are kept separate.
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