ProofAtlas

Minimal surfaces · nonlinear elliptic PDE · real algebraic geometry · exact computation

Polynomial Entire Minimal Graphs

Collaboration beta

Nonlinear smooth entire minimal graphs exist in high dimensions, but can one be the graph of a polynomial? The source has no example and narrows only conditional leading-term branches.

(1+|∇u|2)Δu-∑i,j=1nuiujuij=0,n≥8
Known results and sources
An affine plane contrasted with a ghosted nonlinear graph bearing a question mark, labeled Polynomial Minimal Graphs and Open.
Affine polynomial solutions are known; the nonlinear surface is explicitly a target, not a verified example.

Research problem

Exact mathematical statement

Find an integer n≥8n\ge8 and a non-affine real polynomial u∈ℝ[x1,…,xn]u\in\mathbb R[x_1,…,x_n] satisfying the exact polynomial identity

(1+|∇u|2)Δu-∑i,j=1nuiujuij=0.(1+|\nabla u|^2)\Delta u-\sum_{i,j=1}^n u_i u_j u_{ij}=0.

Approximate residuals and approximate positivity tests are not certificates. The retained source reports no surviving exact non-affine polynomial example.

Problem infographic

Problem at a glance

Four-panel explainer with a flat affine graph, the polynomial minimal-surface equation, the dimension-eight existence question, and a warned hypothetical nonlinear target.
The exact polynomial identity and current boundary: degrees two and three are excluded, while no nonlinear polynomial example is known.

Current mathematical picture

Where work on Polynomial Entire Minimal Graphs stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureRaw full coefficient expansion

The route ledger labels that system unstructured and wasteful and directs work to symmetry-reduced exact modules first. Use multiplicity partitions, harmonic resonance, trace matrices, saturation, and real-singularity tests before coupling.

Route status · Narrowed route
Main reductionConditional degree-19 branch

Conditional on imported dimension-eight geometry, the first arithmetic branch is P19=p^5Q with cubic p and quartic Q.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeIndependently audit the load-bearing local product lemmas, block-radial formula, and two-eigenvalue CAS tables.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Polynomial Entire Minimal Graphs in numbers

2.2kretained lines of mathematical investigation2,242 in the current working snapshot
Argument development
1,970 · 88%
Explored or eliminated routes
56 · 2%
Computational analysis
65 · 3%
Open obligations
41 · 2%
Definitions and setup
110 · 5%
7selected mapped statements2routes investigated5open questions5contribution-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.

Proposed byYifan Guo (modern explicit formulation)Proposed2024Status checkedAug 14, 2026
Source recordedAug 14, 2026

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

14 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

14 selected steps

Scroll horizontally to explore the route

Working route overview for Polynomial Entire Minimal GraphsA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Can a nonlinear polynomial define an entire minimal graph? — Depends on missing premiseCan a nonlinear polynomialdefine an entire minimalgraph?Conditional degree-19 branch — Depends on missing premiseConditional degree-19 branchCurrent reduction — Depends on missing premiseCurrent reductionClosing target — Depends on missing premiseClosing targetExact polynomial target — Depends on missing premiseExact polynomial targetNo surviving certificate — Depends on missing premiseNo surviving certificateUnstructured expansion rejected — Depends on missing premiseUnstructured expansionrejectedRaw full coefficient expansion — stoppedRaw full coefficientexpansionRadial-square-only reducible quartic classification — stoppedRadial-square-only reduciblequartic classificationIndependently audit the load-bearing local product lemmas, block-radial formula, and two-eigenvalue CAS tables. — OpenIndependently audit theload-bearing local productlemmas,…Classify three-or-more-eigenvalue cubic branches for the smallest multiplicity partitions. — OpenClassifythree-or-more-eigenvaluecubic…Classify genuinely angular positive quartics through their complex singular schemes. — OpenClassify genuinely angularpositive quartics throughtheir…First open cubic frontier — OpenFirst open cubic frontierHigher branches remain open — OpenHigher branches remain open
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

2 recorded
Narrowed routeRaw full coefficient expansion

The route ledger labels that system unstructured and wasteful and directs work to symmetry-reduced exact modules first. Use multiplicity partitions, harmonic resonance, trace matrices, saturation, and real-singularity tests before coupling.

Route status · Narrowed route
Narrowed routeRadial-square-only reducible quartic classification

Revision 4 corrects this: any positive-definite quadratic square occurs in the reducible class, including a displayed nonradial fixture. Use the stronger positive-quadratic-cofactor obstruction, which excludes every such square from a leading pair.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Independently audit the load-bearing local product lemmas, block-radial formula, and two-eigenvalue CAS tables.Suggested move: Reproduce each displayed matrix, Gröbner basis, and exceptional-line rejection with exact fixtures and document any mismatch.
Ready to work on
02
Classify three-or-more-eigenvalue cubic branches for the smallest multiplicity partitions.Suggested move: Build exact harmonic-module and trace matrices, saturate pairwise-distinct eigenvalues, and retain positive-semidefinite amplitude data.
Ready to work on
03
First open cubic frontier

Every admissible cubic in the conditional branch must have at least three tangential Hessian eigenvalues at each normalized spherical critical point.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Higher branches remain open

Later dimension-eight branches and all higher dimensions remain live alongside the conditional mainline.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Classify genuinely angular positive quartics through their complex singular schemes.Suggested move: Apply quotient vanishing, Hessian-rank bounds, and the local degree-two through degree-eight hierarchy before broader ansätze.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen problem

Guo’s March 2026 revision says no explicit nonlinear polynomial solution is found. It proves strong restrictions, including nonhomogeneity and degree at least four, but does not construct a solution or prove global nonexistence.

[2]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. PreprintGuo derived a leading-factor structure theorem, ruled out homogeneous solutions and degrees two and three, constrained dimension-eight tangent cones, and still found no explicit polynomial solution.[2]
  2. Peer reviewedBombieri, De Giorgi, and Giusti established the high-dimensional failure of the classical Bernstein conclusion by producing non-affine smooth entire minimal graphs; these are not polynomial examples.[1]
2 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusPolynomial Entire Minimal Graph Problem
Related problemclassical Bernstein problem for smooth entire minimal graphs

Smooth non-affine entire graphs exist in dimensions at least eight, but polynomiality is an additional unresolved algebraic constraint.

[1][2]
Solved special casequadratic and cubic polynomial solutions

The current preprint rules out non-affine polynomial solutions of degrees two and three.

[2]
Dependency or reductiondimension-eight cubic area-minimizing cone

A nonlinear polynomial solution on R^8 would force a cubic area-minimizing but not strictly minimizing tangent cone.

[2]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetFormalize the minimal-surface operator on multivariate real polynomials and equivalence with the graph equation as an exact polynomial identity.
  • Formalization targetFormalize Guo’s leading-factor, tangent-cone, nonhomogeneity, and low-degree exclusion theorems with their analytic and geometric-measure-theory dependencies.
  • Formalization targetDefine a certificate format for algebraic coefficients, exact positivity, and polynomial identity checking before a candidate computation can count as checked evidence.

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 updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
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.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

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.

5 standing statements2 proposed statements5 open questions2 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction2 of 72
  • lemma2 of 72
  • negative result2 of 72
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 statementCan a nonlinear polynomial define an entire minimal graph?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact polynomial targetintermediate
  • retained route statementNo surviving certificateintermediate
  • retained route statementConditional degree-19 branchintermediate
  • retained route statementUnstructured expansion rejectedintermediate
  • 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
  • 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 failureRaw full coefficient expansionreported failure
  • Useful failureRadial-square-only reducible quartic classificationreported failure
  • Research targetIndependently audit the load-bearing local product lemmas, block-radial formula, and two-eigenvalue CAS tables.open
  • Research targetClassify three-or-more-eigenvalue cubic branches for the smallest multiplicity partitions.open
  • Research targetClassify genuinely angular positive quartics through their complex singular schemes.open
  • Research targetFirst open cubic frontieropen
  • Research targetHigher branches remain openopen
  • ComputationThe source specifies exact CAS tasks for harmonic modules, resonance matrices, saturation, factorization, and real singularity, but supplies no surviving certificate.Scalar and two-eigenvalue cubic branches are source-reported closed after targeted audits; the first open cubic frontier has at least three eigenvalues. · reported unreproduced
  • Narrowed routeRaw full coefficient expansionThe route ledger labels that system unstructured and wasteful and directs work to symmetry-reduced exact modules first. Use multiplicity partitions, harmonic resonance, trace matrices, saturation, and real-singularity tests before coupling.
  • Narrowed routeRadial-square-only reducible quartic classificationRevision 4 corrects this: any positive-definite quadratic square occurs in the reducible class, including a displayed nonradial fixture. Use the stronger positive-quadratic-cofactor obstruction, which excludes every such square from a leading pair.
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

Priority open bridgeIndependently audit the load-bearing local product lemmas, block-radial formula, and two-eigenvalue CAS tables.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

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.

Read-only beta · actions unavailable
Prepared starting pointIndependently audit the load-bearing local product lemmas, block-radial formula, and two-eigenvalue CAS tables.

Polynomial Entire Minimal Graphs · ready to start

Mathematical updatesFollow this problem

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

Nonlinear smooth entire minimal graphs exist in high dimensions, but can one be the graph of a polynomial? The source has no example and narrows only conditional leading-term branches.

  • 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
Hosted agentRun this task with a hosted agent

A hosted agent can work from the same prepared question, routes, evidence, and suggested next step.

Your own AI agentConnect an outside research agent

Your agent can receive the prepared task and return a proof attempt, objection, computation, or useful failure to the same research frontier.

Sources and references2 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Minimal cones and the Bernstein problempeer reviewed result · Enrico Bombieri, Ennio De Giorgi, Enrico Giusti · Inventiones mathematicae 7, 243-268 · 1969 · accessed Aug 14, 2026
  2. 2
    On polynomial solutions to the minimal surface equationpreprint · Yifan Guo · arXiv; to appear in Calculus of Variations and Partial Differential Equations · 2024; revised 2026-03-16 · ARXIV 2404.00115 · accessed Aug 14, 2026

Important qualifications

  • The modern construction problem is sourced to Guo’s 2024 preprint, revised in 2026 and listed as forthcoming; this record does not infer a longer historical named-conjecture lineage.
  • The classical Bombieri–De Giorgi–Giusti result supplies non-affine smooth entire minimal graphs in high dimension, not polynomial examples.
  • No external computation or packet attachment was run, and no exact formalization of the construction problem was located in bounded official-repository searches.
  • External metadata is independent of the private packet and grants no proof, review, credit, publication, or rights authority.

Continue exploring

Compare another research frontier

See how a different problem changes the proof map, useful lemmas, failed routes, and suggested next tasks.

Explore all research workspaces

Expanded visual

Open original image