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Diophantine geometry · undecidability · rational points

Hilbert’s Tenth Problem over Q

Collaboration beta

Given any polynomial equation with integer coefficients, can an algorithm always determine whether it has a rational solution? The answer over Q remains unknown. The retained handoff proposes an undecidability reduction but explicitly leaves its decisive normalization lemma open.

∃algorithm deciding{f∈ℤ[x]:∃q∈ℚn, f(q)=0} ?
Listed inAIM Extensions of Hilbert’s Tenth Problem problem list
Known results and sources
Editorial H10(Q) card showing an arbitrary integer polynomial, the rational-solution decision question, and an open status without asserting any example outcome.
Hilbert’s Tenth Problem over Q asks whether one algorithm can decide rational solvability for every integer polynomial; it remains open.

Research problem

Exact mathematical statement

Does there exist an algorithm that, given any polynomial

f∈ℤ[x1,…,xn],f\in\mathbb Z[x_1,…,x_n],

decides whether there is a tuple (q1,…,qn)∈ℚn(q_1,…,q_n)\in\mathbb Q^n with f(q1,…,qn)=0f(q_1,…,q_n)=0? The problem is open; neither decidability nor undecidability over ℚ\mathbb Q is asserted here.

Problem infographic

Problem at a glance

Problem-first explainer for H10(Q), contrasting the solved undecidability result over integers with the open rational problem and isolating the source’s missing arithmetic-normalization relation.
The source proposes an elliptic undecidability reduction but explicitly stops at a missing fixed existential normalization relation; finite script checks do not prove that open step.

Current mathematical picture

Where work on Hilbert’s Tenth Problem over Q stands

Open problem

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

Useful failureHomogeneous scale normalization

The source records a quadratic gauge and non-torsion cocycle that preserve or obstruct the attempted normalizations. Paired complementary localizations, reciprocal-scale coupling, or a special Cayley–Bézout primitive-row construction remain explicitly open routes.

Route status · Narrowed route
Main reductionBranch-robust M=2 window

Conditional on a sound generator of the actual branch ideal, the source states xy=z iff the generator norm lies in the 2^±36 window.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeBuild a fixed existential generator relation.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

Hilbert’s Tenth Problem over Q in numbers

3.2kretained lines of mathematical investigation3,243 in the current working snapshot
Argument development
2,819 · 87%
Explored or eliminated routes
52 · 2%
Computational analysis
104 · 3%
Open obligations
89 · 3%
Definitions and setup
179 · 6%
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 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

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 Hilbert’s Tenth Problem over QA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Can rational solvability be decided uniformly? — Depends on missing premiseCan rational solvability bedecided uniformly?Branch-robust M=2 window — Depends on missing premiseBranch-robust M=2 windowCurrent reduction — Depends on missing premiseCurrent reductionSharper M=18 interface — Depends on missing premiseSharper M=18 interfaceClosing target — Depends on missing premiseClosing targetExplicit elliptic integer copy — Depends on missing premiseExplicit elliptic integercopyVerification scripts have bounded scope — Depends on missing premiseVerification scripts havebounded scopeHomogeneous scale normalization — stoppedHomogeneous scalenormalizationBuild a fixed existential generator relation. — OpenBuild a fixed existentialgenerator relation.Close a reciprocal-scale or localization route. — OpenClose a reciprocal-scale orlocalization route.Audit the full reduction after normalization. — OpenAudit the full reductionafter normalization.
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 routeHomogeneous scale normalization

The source records a quadratic gauge and non-torsion cocycle that preserve or obstruct the attempted normalizations. Paired complementary localizations, reciprocal-scale coupling, or a special Cayley–Bézout primitive-row construction remain explicitly open routes.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Build a fixed existential generator relation.Suggested move: Express the actual branch-dependent M=2 ideal generator by a bounded existential formula and prove every-witness norm soundness.
Ready to work on
02
Close a reciprocal-scale or localization route.Suggested move: Construct the missing opposite-character bridge or uniform two-chart generator certificate without informal integrality or unbounded support conditions.
Ready to work on
03
Audit the full reduction after normalization.Suggested move: Once a Tier-C lemma exists, verify sign, branch, valuation, first-order-language, and MRDP dependencies before asserting undecidability.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen problem

The ordinary problem of deciding whether arbitrary algebraic varieties over Q have rational points remains open. A 2023 primary preprint proves undecidability only after adjoining finitely many height-comparison conditions, not for ordinary H10(Q).

[1][2]
External progress

What the literature has established

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

  1. PreprintGarcia-Fritz, Pasten, and Vidaux proved undecidability for rational-point existence augmented by finitely many height-comparison conditions while explicitly retaining ordinary H10(Q) as open.[2]
  2. Authoritative summaryAn American Institute of Mathematics workshop organized open problems around extensions of Hilbert’s Tenth Problem, including the rational case.[1]
2 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusHilbert’s Tenth Problem over Q
Solved special caseHilbert’s Tenth Problem over Z

Hilbert’s Tenth Problem over the integers is undecidable by the Davis–Putnam–Robinson–Matiyasevich theorem; changing the ground field to Q creates the unresolved variant.

[1]
Weaker or relaxed formRational-point existence with height comparisons

Adding finitely many height-comparison conditions yields a proved undecidability result, but those extra conditions are not part of ordinary H10(Q).

[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 of uniform rational-solvability decision for arbitrary integer polynomials.
  • Formalization targetFormal Diophantine definability infrastructure over Q and the exact logical reduction from integer multiplication.
  • Formalization targetFormal separation of ordinary existential polynomial equations from auxiliary height-comparison predicates.

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
  • reduction3 of 73
  • lemma2 of 72
  • computational claim1 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 statementCan rational solvability be decided uniformly?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExplicit elliptic integer copyintermediate
  • retained route statementBranch-robust M=2 windowintermediate
  • retained route statementSharper M=18 interfaceintermediate
  • retained route statementVerification scripts have bounded scopeintermediate
  • 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 failureHomogeneous scale normalizationreported failure
  • Research targetBuild a fixed existential generator relation.open
  • Research targetClose a reciprocal-scale or localization route.open
  • Research targetAudit the full reduction after normalization.open
  • Research targetNo complete proofsuperseded
  • Research targetArithmetic normalization remains opensuperseded
  • Narrowed routeHomogeneous scale normalizationThe source records a quadratic gauge and non-torsion cocycle that preserve or obstruct the attempted normalizations. Paired complementary localizations, reciprocal-scale coupling, or a special Cayley–Bézout primitive-row construction remain explicitly open routes.
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 bridgeBuild a fixed existential generator relation.

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

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Prepared starting pointBuild a fixed existential generator relation.

Hilbert’s Tenth Problem over Q · 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

Given any polynomial equation with integer coefficients, can an algorithm always determine whether it has a rational solution? The answer over Q remains unknown. The retained source material proposes an undecidability reduction but explicitly leaves its decisive normalization lemma 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
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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
    Extensions of Hilbert’s Tenth Problemmaintained problem list · American Institute of Mathematics · accessed Aug 14, 2026
  2. 2
    Effectivity for existence of rational points is undecidablepreprint · Natalia Garcia-Fritz, Hector Pasten, Xavier Vidaux · arXiv · 2023-11-03 · ARXIV 2311.01958 · accessed Aug 14, 2026

Important qualifications

  • The collection distinguishes ordinary rational solvability from variants with added height-comparison predicates.
  • The AIM page is a maintained workshop problem list rather than a current exhaustive bibliography.
  • No claim is made that absence of a fuller formalization or later related result was exhaustively established.

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