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Complex analysis · polynomial critical points · extremal problems

Smale's Mean Value Problem

Collaboration beta

Exact centered cases and low-support algebra are strong, yet the sharp inequality for arbitrary complex polynomials remains open.

minP'(ζ)=0|P(ζ)-P(z)(ζ-z)P'(z)|≤1-1n
Known results and sources
Problem-first open-research thumbnail for Smale's Mean Value Problem, showing an unresolved mathematical structure without a completion mark or navigation arrow.
How close must a critical value be? The page presents source-reported partial structure without claiming a proof.

Research problem

Exact mathematical statement

For every complex polynomial of degree n≥2 and every noncritical base point, prove the sharp mean-value bound with constant 1−1/n.

minP'(ζ)=0|P(ζ)-P(z)(ζ-z)P'(z)|≤1-1n\min_{P'(\zeta)=0}\left|\frac{P(\zeta)-P(z)}{(\zeta-z)P'(z)}\right|\le1-\frac1n

This remains open; the source reports partial results and route constraints, not a complete proof.

Problem infographic

Problem at a glance

Three-panel source-bound explainer for Smale's Mean Value Problem: exact target, strongest retained result, and the unresolved closing bridge.
The source separates the exact target, retained partial results, and the still-open bridge.

Current mathematical picture

Where work on Smale's Mean Value Problem stands

Open problem

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

Useful failureFull phase erasure

The (6,6,3) phase-erased fixed-defect-three moment is negative. Phase-refined envelopes, determinant conjugation, cycle inequalities, and support compression remain viable.

Route status · Narrowed route
Main reductionCurrent reduction

The active global route is to prove the exact confluent 3×3 determinant inequality or reduce every global maximizer by support compression or collision to an already solved support level.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeResolve the first (6,5,4) asymmetric pattern.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

Smale's Mean Value Problem in numbers

1.9kretained lines of mathematical investigation1,918 in the current working snapshot
Argument development
1,644 · 86%
Explored or eliminated routes
47 · 2%
Computational analysis
91 · 5%
Open obligations
53 · 3%
Definitions and setup
83 · 4%
7selected mapped statements1routes investigated3open questions3contribution-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.

Status checkedAug 21, 2026
Source recordedAug 21, 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

11 selected steps

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

11 selected steps

Scroll horizontally to explore the route

Working route overview for Smale's Mean Value ProblemA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.How close must a critical value be? — Depends on missing premiseHow close must a criticalvalue be?Current reduction — Depends on missing premiseCurrent reductionSharp mean-value target — Depends on missing premiseSharp mean-value targetCentered degree-14 frontier — Depends on missing premiseCentered degree-14 frontierClosing target — Depends on missing premiseClosing targetPhase-erasure obstruction — Depends on missing premisePhase-erasure obstructionTwo-support theorem — Depends on missing premiseTwo-support theoremFull phase erasure — stoppedFull phase erasureResolve the first (6,5,4) asymmetric pattern. — OpenResolve the first (6,5,4)asymmetric pattern.Prove a phase-sensitive determinant inequality. — OpenProve a phase-sensitivedeterminant inequality.Establish global support compression or full stratified KKT. — OpenEstablish global supportcompression or fullstratified…
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

1 recorded
Narrowed routeFull phase erasure

The (6,6,3) phase-erased fixed-defect-three moment is negative. Phase-refined envelopes, determinant conjugation, cycle inequalities, and support compression remain viable.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Resolve the first (6,5,4) asymmetric pattern.Suggested move: Build a multiplicity-sensitive centered selector.
Ready to work on
02
Prove a phase-sensitive determinant inequality.Suggested move: Seek a diagonal conjugation or paired-cycle identity for the centered determinant.
Ready to work on
03
Establish global support compression or full stratified KKT.Suggested move: Prove support compression to at most three critical locations or develop the full stratified KKT theory.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 21, 2026
Current statusOpen problem

Smale's sharp mean-value problem remains open in general; the literature contains lower-degree, normalized, and structural partial results.

[1][2]
External progress

What the literature has established

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

  1. Peer reviewedSmale's mean value conjecture and the coefficients of univalent functions supplies a representative external result or boundary relevant to the problem; it is not treated here as a proof of the full packet target.[2]
  2. Historical sourceThe fundamental theorem of algebra and complexity theory supplies a representative external result or boundary relevant to the problem; it is not treated here as a proof of the full packet target.[1]
2 cited sources1 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusSmale's Mean Value Problem
Related problemSmale's Mean Value Problem

Smale's sharp mean-value problem remains open in general; the literature contains lower-degree, normalized, and structural partial results.

[1][2]

Formalization opportunities

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

  • Formalization targetA formal statement matching the exact public target and all quantifiers.
  • Formalization targetFormal libraries for the principal mathematical structures used by the strongest route.
  • Formalization targetA checked closing argument for the source-identified open bridge.

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 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

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 statements3 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction1 of 71
  • lemma3 of 73
  • equivalence1 of 71
  • negative result1 of 71
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
  • retained route statementHow close must a critical value be?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementSharp mean-value targetintermediate
  • retained route statementTwo-support theoremintermediate
  • retained route statementCentered degree-14 frontierintermediate
  • retained route statementPhase-erasure obstructionintermediate
  • 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 failureFull phase erasurereported failure
  • Research targetResolve the first (6,5,4) asymmetric pattern.open
  • Research targetProve a phase-sensitive determinant inequality.open
  • Research targetEstablish global support compression or full stratified KKT.open
  • Research targetMultiplicity-sensitive selectorsuperseded
  • Research targetGlobal support bridgesuperseded
  • Narrowed routeFull phase erasureThe (6,6,3) phase-erased fixed-defect-three moment is negative. Phase-refined envelopes, determinant conjugation, cycle inequalities, and support compression remain viable.
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 bridgeResolve the first (6,5,4) asymmetric pattern.

1 approach has 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 pointResolve the first (6,5,4) asymmetric pattern.

Smale's Mean Value Problem · 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

Exact centered cases and low-support algebra are strong, yet the sharp inequality for arbitrary complex polynomials remains open.

  • 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 21, 2026

The mathematical context was checked on Aug 21, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    The fundamental theorem of algebra and complexity theoryoriginal source · Steve Smale · Bulletin of the AMS · 1981 · accessed Aug 21, 2026
  2. 2
    Smale's mean value conjecture and the coefficients of univalent functionspeer reviewed result · Anthony Conte, Ege Fujikawa, Nikola Lakic · Proceedings of the AMS · 2007 · accessed Aug 21, 2026

Important qualifications

  • This was a bounded status and identity check, not an exhaustive bibliography, priority review, or legal review.
  • Private packet claims were not treated as external authority; submitted links and attachments were not executed or actively rendered.
  • No absence claim is inferred from the bounded search, and recent preprints remain subject to ordinary scholarly review.

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