Diophantine geometry · heights · algebraic points of bounded degree · normal-crossings divisors · arithmetic discriminants

Vojta’s Conjecture over Number Fields

Collaboration beta

Can proximity to a normal-crossings divisor plus canonical height be controlled by one field discriminant and an arbitrarily small big-height allowance for every algebraic point of bounded degree outside a proper exceptional set?

m_{D, S} (P) + h_{K_{X}} (P) \leq d_{k} (P) + ε h_{A} (P) + O (1), [k (P) : k] \leq r

Known results and sources

Vojta's bounded-degree number-field conjecture relates divisor proximity, canonical height, one field discriminant, and a small big-height allowance; the unfinished geometry signals that the general inequality remains open.

Research problem

Exact mathematical statement

Let k be a number field, X/k a smooth projective variety, D a reduced simple-normal-crossings divisor, A a big divisor, S a finite set of places of k, r ≥ 1, and ε > 0. For an algebraic point P, let d_k(P) denote the normalized logarithmic discriminant of k(P)/k. The bounded-degree form predicts a proper closed subset Z of X such that

m_{D,S}(P)+h_{K_X}(P) ≤ d_k(P)+ε h_A(P)+O(1)

for every P outside Z with [k(P):k] ≤ r. The retained source explicitly reports no proof. Its fully truncated inequality is a stronger working target and is not presented here as already equivalent to the conjecture.

Problem infographic

Problem at a glance

Problem-first landscape explainer for the bounded-degree Vojta inequality: a number field and smooth projective variety with a reduced normal-crossings divisor, algebraic points of degree at most r, a balance comparing proximity plus canonical height with one field discriminant plus epsilon-height allowance, a local crossing that distinguishes total-union counting from componentwise counting, and an explicit open-over-number-fields boundary. — The conjecture asks for one discriminant to control bounded-degree proximity and canonical height outside a proper exceptional set. Total truncation is a stronger working target, and function-field analogues or selected special cases do not prove the general number-field statement.

Current mathematical picture

Where work on Vojta’s Conjecture over Number Fields stands

Open conjecture

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

Useful failureFull multi-Kummer saturation failure

In the tame a=b=1 example with t(P)=u(P)=π and θ^ℓ=π, the product suborder R[θ²] satisfies 2 length(R[θ]/R[θ²])=ℓ−1, while adjoining either individual parameter root already gives the normal order R[θ], so the full individual-root order has normalization index zero. Construct or rule out a non-tautological mixed determinant that couples one total-boundary cyclic Kummer order to additive Plücker relations, retains one field discriminant, and reaches coefficient one without…

Route status · Narrowed route

Main reductionCurrent reduction

The source proposes that suitable projective-radical or relative moving-target theorems would imply broader Vojta inequalities. It uses the degree-five del Pezzo pair as a laboratory for the first varying marked-line case, while explicitly leaving the relative all-dimensional chain and uniformity requirements incomplete.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve the local cyclic-total versus multi-Kummer order comparison.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

Vojta’s Conjecture over Number Fields in numbers

734retained lines of mathematical investigation734 in the current working snapshot

Argument development: 641 · 87%
Explored or eliminated routes: 24 · 3%
Computational analysis: 11 · 1%
Open obligations: 15 · 2%
Definitions and setup: 43 · 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.

Proposed byPaul VojtaProposed1987Status checkedAug 13, 2026

Source recordedAug 13, 2026

Recommended next task

Prove the local cyclic-total versus multi-Kummer order comparison.

Suggested move: Work in a tame DVR with theta^ell equal to a uniformizer, include unit twists, and compute the normalization indices of the product suborder and the order generated by individual powers via numerical semigroups or Apéry sets; recover the a=b=1 saturation counterexample exactly.

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

14 selected steps

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

2 recorded

Narrowed routeFull multi-Kummer saturation failure

Route status · Narrowed route

Narrowed routeValue-only coefficient barrier

The exact source-reported ratio of total line-bundle height degree to the best divisorial filtration gain tends to 15/13, which is strictly larger than coefficient one. A mixed determinant in which Kummer conjugate rows and pentagon/Cox section columns interact through additive circuit relations, first tested on the three-point projective line.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks

Prove the local cyclic-total versus multi-Kummer order comparison.Suggested move: Work in a tame DVR with theta^ell equal to a uniformizer, include unit twists, and compute the normalization indices of the product suborder and the order generated by individual powers via numerical semigroups or Apéry sets; recover the a=b=1 saturation counterexample exactly.

Ready to work on

Globalize one total-boundary cyclic order without component-root saturation.Suggested move: On the degree-five del Pezzo surface, choose and audit either one cyclic cover of an enlarged total boundary or the total-boundary root stack; compute normalization, crossing singularities, conductor terms, log Riemann–Hurwitz, auxiliary-height growth, and bounded-degree effects.

Ready to work on

Bounded-degree height target

Establish the displayed bounded-degree Vojta inequality with one normalized discriminant outside one proper closed exceptional subset.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Produce or falsify a coefficient-one Kummer–Plücker determinant.Suggested move: Build a determinant whose Kummer conjugate rows and pentagon/Cox section columns interact through an additive circuit, verify a finite-place divisibility term not forced by the scalar discriminant identity, and audit the exact archimedean coefficient first on P1 minus {0,1,infinity}.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 13, 2026

Current statusOpen conjecture

The bounded-degree number-field conjecture remains open in its general smooth-projective-pair form. The curve-level number-field 1+epsilon problem already contains abc-level difficulty. Strong function-field analogues, Nevanlinna counterparts, and selected rational-surface theorems do not establish the universal number-field statement.

[2][8][7]

External progress

What the literature has established

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

2024
Authoritative summaryA contemporary survey continued to treat Vojta's higher-dimensional abc/abcd framework as conjectural while explaining conditional consequences in arithmetic dynamics.[8]
2018
Peer reviewedYasufuku proved Vojta-type statements for some rational surfaces and established conditional relationships with abc for other explicit surfaces, without resolving the general bounded-degree conjecture.[7]
2004–2009
Peer reviewedWork of McQuillan and Yamanoi established strong function-field analogues of the curve-level 1+epsilon problem; the base-field change is essential and does not settle the number-field conjecture.[6][2]
2000
Peer reviewedVojta proved that a weaker rational-point Diophantine-approximation conjecture with an additional group-action hypothesis implies abc, and proved analogues only in the split function-field and holomorphic…[5]

Mathematical neighborhood

Related results and reusable starting points

Current focusVojta's conjecture for algebraic points of bounded degree over number fields

Stronger or generalized formVojta's more general abc conjecture

Vojta's truncated general abc conjecture extends both the classical abc conjecture and his bounded-degree Diophantine conjecture. Its total truncated counting term is a stronger target and must not be silently substituted for the ordinary bounded-degree statement.

[4]

Logical consequenceMasser–Oesterlé abc conjecture

Suitable Vojta-type height inequalities imply the classical abc conjecture, while separate curve-level results explain converse implications under their own formulations. This does not identify the full higher-dimensional bounded-degree statement with abc.

[5][2]

Solved special caseStrong abc / Vojta 1+epsilon over function fields

The strong curve-level analogue is known over characteristic-zero function fields after McQuillan and Yamanoi. It is a different arithmetic setting from number fields.

[6][2]

Solved special caseSelected rational-surface cases

Some explicitly described rational surfaces satisfy scoped Vojta statements, and related surfaces are conditionally linked to abc. These results do not supply the universal smooth-projective-pair theorem.

[7]

Related problemNevanlinna Second Main Theorem analogues

The conjectural arithmetic inequalities are organized by a precise dictionary with Nevanlinna value-distribution theory. Analytic analogues motivate the statement but are not number-field proofs.

[1][2]

Formalization opportunities

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

Formalization targetA proof-assistant development would need a pinned theory of number fields, normalized local/global heights, Weil functions and proximity functions, big and canonical divisors, simple-normal-crossings support, algebraic points with bounded residue-field degree, and normalized logarithmic discriminants.
Formalization targetThe formal target must distinguish the ordinary counting form from the stronger total-truncated and componentwise-truncated variants, including the exceptional closed subset and every dependence of constants.
Formalization targetAny formalized special case or implication must bind its exact variety, divisor, point degree, place set, height normalization, discriminant convention, and exceptional-set scope before it can be compared with this workspace.

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 statements3 proposed statements4 open questions2 narrowed routes

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
reduction2 of 82
lemma2 of 82
negative result2 of 82
counterexample1 of 81

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 statementDoes one discriminant control bounded-degree proximity and canonical height?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementTotal and componentwise truncation differintermediate
retained route statementCount–different comparisonintermediate
retained route statementValue-only coefficient barrierintermediate
retained route statementFull multi-Kummer saturation obstructionintermediate
retained route statementTotal-boundary mixed-determinant frontierintermediate
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 counterexample narrows one intermediate strategy; it does not challenge the open conjecture.challenges · 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 multi-Kummer saturation failurereported failure
Useful failureValue-only coefficient barrierreported failure
Research targetProve the local cyclic-total versus multi-Kummer order comparison.open
Research targetGlobalize one total-boundary cyclic order without component-root saturation.open
Research targetProduce or falsify a coefficient-one Kummer–Plücker determinant.open
Research targetBounded-degree height targetopen
Narrowed routeFull multi-Kummer saturation failureIn the tame a=b=1 example with t(P)=u(P)=π and θ^ℓ=π, the product suborder R[θ²] satisfies 2 length(R[θ]/R[θ²])=ℓ−1, while adjoining either individual parameter root already gives the normal order R[θ], so the full individual-root order has normalization index zero. Construct or rule out a non-tautological mixed determinant that couples one total-boundary cyclic Kummer order to additive Plücker relations, retains one field discriminant, and reaches coefficient one without losing bounded-degree or exceptional-set control.
Narrowed routeValue-only coefficient barrierThe exact source-reported ratio of total line-bundle height degree to the best divisorial filtration gain tends to 15/13, which is strictly larger than coefficient one. A mixed determinant in which Kummer conjugate rows and pentagon/Cox section columns interact through additive circuit relations, first tested on the three-point projective line.

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 bridgeProve the local cyclic-total versus multi-Kummer order comparison.

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.

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Survive an independent attempt to falsify the proposed step.

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Prepared starting pointProve the local cyclic-total versus multi-Kummer order comparison.

Vojta’s Conjecture over Number Fields · ready to start

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Sources and references8 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.

1
Diophantine Approximations and Value Distribution Theoryoriginal source · Paul Vojta · Springer, Lecture Notes in Mathematics 1239 · 1987 · DOI 10.1007/BFb0072989 · accessed Aug 13, 2026
2
Diophantine Approximation and Nevanlinnasurvey or monograph · Paul Vojta · Paul Vojta, University of California, Berkeley · 2007/2008 lecture notes · accessed Aug 13, 2026
3
On algebraic points on curvespeer reviewed result · Paul Vojta · Compositio Mathematica 78, 29–36 · 1991 · accessed Aug 13, 2026
4
A more general abc conjecturepeer reviewed result · Paul Vojta · International Mathematics Research Notices 1998(21), 1103–1116 · 1998 · ARXIV math/9806171 · accessed Aug 13, 2026
5
On the abc conjecture and diophantine approximation by rational pointspeer reviewed result · Paul Vojta · American Journal of Mathematics 122(4), 843–872 · 2000 · ARXIV math/9908024 · accessed Aug 13, 2026
6
The Strong abc conjecture over function fields [after McQuillan and Yamanoi]survey or monograph · Carlo Gasbarri · Société Mathématique de France, Astérisque 326, Exposé 989 · 2009 · MR MR2605324 · ZBMATH Zbl 1190.14023 · accessed Aug 13, 2026
7
Vojta's conjecture on rational surfaces and the abc conjecturepeer reviewed result · Yu Yasufuku · Forum Mathematicum 30(3), 631–649 · 2018-05-01 · DOI 10.1515/forum-2017-0089 · accessed Aug 13, 2026
8
The abcd conjecture, uniform boundedness, and dynamical systemssurvey or monograph · Robin Zhang · Publications mathématiques de Besançon. Algèbre et théorie des nombres · 2024-04-22 · ARXIV 2206.09725 · DOI 10.5802/pmb.58 · accessed Aug 13, 2026

Important qualifications

This record is source-separated administrative context. It does not use the unreviewed source material as external evidence and grants no mathematical, rights, review, publication, or deployment authority.
The title is deliberately scoped to algebraic points of bounded degree over number fields. Rational-point, curve-only, truncated-abc, function-field, and Nevanlinna formulations are related but not interchangeable statements.
The 1987 monograph and Vojta's later lecture notes support the historical identity and conjectural framework; neither is evidence that the unreviewed source material's reductions or determinant calculations are correct.
Function-field results after McQuillan and Yamanoi and selected rational-surface theorems are neighboring results. They do not prove the general number-field conjecture recorded here.
Scoped searches of public Lean and mathlib surfaces did not locate a checked formalization of the exact bounded-degree number-field inequality. This does not establish nonexistence elsewhere.
The status review did not attempt to adjudicate disputed claims surrounding the separate abc conjecture. This record relies only on sources that continue to present the relevant Vojta statements as conjectural and on explicitly scoped theorems.
No computation, code, dataset, or finite certificate can by itself establish this universal height inequality; no such resource is promoted as theorem evidence.

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