ProofAtlas

Arithmetic dynamics and Diophantine geometry

Morton–Silverman Uniform Boundedness Conjecture

Collaboration beta

For fixed dimension, map degree, and number-field degree, should one bound control every rational preperiodic point of every such dynamical system? The conjecture remains open, including the exact-period obstruction for quadratic polynomials.

#PrePer(f,K)≤B(N,d,D)
Known results and sources
Dark collection cover showing a projective morphism over a bounded-degree number field, finite tails and cycles of rational points, the exact uniform-preperiodic-point question, and open status.
For fixed projective dimension, morphism degree and field-degree bound, the conjecture asks for one bound on all K-rational preperiodic points.

Research problem

Exact mathematical statement

For fixed integers N≥1N\ge1, d≥2d\ge2, and D≥1D\ge1, does there exist a constant B(N,d,D)B(N,d,D) such that every degree-dd morphism f:PKN→PKNf: \mathbf P^N_K\to\mathbf P^N_K over every number field KK with [K:Q]≤D[K: \mathbf Q]\le D satisfies

#PrePer(f,K)≤B(N,d,D)?\#\operatorname{PrePer}(f,K)\le B(N,d,D)?

Here PrePer(f,K)\operatorname{PrePer}(f,K) denotes the KK-rational points with finite forward orbit under ff.

Problem infographic

Problem at a glance

Problem-first Morton–Silverman explainer defining projective morphisms over bounded-degree number fields and finite rational orbits, stating the exact uniform bound question, and separating a published restricted periodic-point theorem from the fully open preperiodic conjecture.
A 2026 peer-reviewed theorem bounds periodic points for a restricted good-reduction polynomial class; the uniform bound for all K-rational preperiodic points of arbitrary projective morphisms remains open.

Current mathematical picture

Where work on Morton–Silverman Uniform Boundedness Conjecture stands

Open conjecture

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

Useful failurePurely local period bound

At sufficiently deep bad 2-adic places, both inverse branches contract and every binary word occurs as a rational periodic orbit. Global adelic, semiabelian, or moving-S-unit methods remain viable because they can couple valuation data across places.

Route status · Narrowed route
Main reductionConditional fixed-period tail reduction

For every fixed quadratic exact period, the source reports a uniform tail bound only subject to its stated transversality and audit dependencies.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeProve moving-cusp adelic rigidity for bounded-degree exact quadratic cycles outside explicitly classified special loci.Task status · Ready to work on

Work mapped so far

Morton–Silverman Uniform Boundedness Conjecture in numbers

2.7kretained lines of mathematical investigation2,691 in the current working snapshot
Argument development
2,371 · 88%
Explored or eliminated routes
30 · 1%
Computational analysis
13 · 0%
Open obligations
69 · 3%
Definitions and setup
208 · 8%
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 byPatrick Morton and Joseph H. SilvermanProposed1994Status checkedAug 15, 2026
Source recordedAug 15, 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 Morton–Silverman Uniform Boundedness ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Is #PrePer(f,K) uniformly bounded by N, d, and D alone? — Depends on missing premiseIs #PrePer(f,K) uniformlybounded by N, d, and Dalone?Conditional fixed-period tail reduction — Depends on missing premiseConditional fixed-periodtail reductionCurrent reduction — Depends on missing premiseCurrent reductionClosing target — Depends on missing premiseClosing targetConditional low-period tail bounds — Depends on missing premiseConditional low-period tailboundsCusp labels move with the prime — Depends on missing premiseCusp labels move with theprimeLocal full shift blocks local-only proofs — Depends on missing premiseLocal full shift blockslocal-only proofsPurely local period bound — stoppedPurely local period boundBoundary convexity alone — stoppedBoundary convexity aloneProve moving-cusp adelic rigidity for bounded-degree exact quadratic cycles outside explicitly classified special loci. — OpenProve moving-cusp adelicrigidity for bounded-degreeexact…Find a label-independent quotient that controls all moving cuspidal differences while retaining at least 2D cotangent directions. — OpenFind a label-independentquotient that controls allmoving…Control both moving cuspidal differences and the arithmetic kernel of a primitive semiabelian quotient uniformly in the period. — OpenControl both moving cuspidaldifferences and thearithmetic…Exact global boundedness target — OpenExact global boundednesstargetExact periods remain the primary open module — OpenExact periods remain theprimary open module
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 routePurely local period bound

At sufficiently deep bad 2-adic places, both inverse branches contract and every binary word occurs as a rational periodic orbit. Global adelic, semiabelian, or moving-S-unit methods remain viable because they can couple valuation data across places.

Route status · Narrowed route
Narrowed routeBoundary convexity alone

Different binary cusps provide balancing rays, so a proof must also use nonlinear cycle equations, arithmetic quotients, or classification. A combined logarithmic and ordinary formal-immersion argument remains a candidate after the cusp-difference subgroup is controlled.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Prove moving-cusp adelic rigidity for bounded-degree exact quadratic cycles outside explicitly classified special loci.Suggested move: Construct the full exact cusp boundary and compute the image of general exact-word cusps in the augmentation quotient.
Ready to work on
02
Find a label-independent quotient that controls all moving cuspidal differences while retaining at least 2D cotangent directions.Suggested move: Determine the image of every low-degree exact cusp difference in a primitive semiabelian quotient.
Ready to work on
03
Exact global boundedness target

The conjecture quantifies over every number field of degree at most D and every degree-d morphism of projective N-space.

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

The source audit lists uniform exact periods, uniform quadratic preperiodic counts, cuspidal-difference control, and the full Morton–Silverman statement as open.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Control both moving cuspidal differences and the arithmetic kernel of a primitive semiabelian quotient uniformly in the period.Suggested move: Test whether moving cuspidal differences and the arithmetic specialization kernel are uniformly controlled in one primitive quotient.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 15, 2026
Current statusOpen conjecture

The full Morton–Silverman conjecture for all K-rational preperiodic points of arbitrary degree-d morphisms of projective N-space remains open. A recent theorem settles a restricted good-reduction polynomial class for periodic points only.

[2][3]
External progress

What the literature has established

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

  1. Peer reviewedRajagopal and Zhang publish an explicit periodic-point bound for a restricted good-reduction polynomial class, including certain unicritical maps.[3]
  2. Peer reviewedA broad Bulletin of the AMS survey states the full projective-space uniform boundedness conjecture and reviews limited progress.[2]
  3. Historical sourceMorton and Silverman formulated the uniform boundedness problem for rational periodic and preperiodic dynamics over number fields.[1]
3 cited sources1 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusMorton–Silverman Uniform Boundedness Conjecture
Solved special caseGood-reduction polynomial periodic points

For a restricted good-reduction polynomial class, the number of K-rational periodic points is at most d^[K:Q]; this does not bound all preperiodic points for every morphism.

[3]

Formalization opportunities

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

  • Formalization targetFormal arithmetic dynamics of morphisms on projective space over number fields.
  • Formalization targetFormal height, good-reduction, local-period, and rational preperiodic-point finiteness theorems.
  • Formalization targetA formal uniform exact-period theorem or another route controlling all cycles and tails.

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.22 displayed rows · 2 routes included
  • retained route statementIs #PrePer(f,K) uniformly bounded by N, d, and D alone?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementConditional low-period tail boundsintermediate
  • retained route statementConditional fixed-period tail reductionintermediate
  • retained route statementLocal full shift blocks local-only proofsintermediate
  • retained route statementCusp labels move with the primeintermediate
  • 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 failurePurely local period boundreported failure
  • Useful failureBoundary convexity alonereported failure
  • Research targetProve moving-cusp adelic rigidity for bounded-degree exact quadratic cycles outside explicitly classified special loci.open
  • Research targetFind a label-independent quotient that controls all moving cuspidal differences while retaining at least 2D cotangent directions.open
  • Research targetControl both moving cuspidal differences and the arithmetic kernel of a primitive semiabelian quotient uniformly in the period.open
  • Research targetExact global boundedness targetopen
  • Research targetExact periods remain the primary open moduleopen
  • Narrowed routePurely local period boundAt sufficiently deep bad 2-adic places, both inverse branches contract and every binary word occurs as a rational periodic orbit. Global adelic, semiabelian, or moving-S-unit methods remain viable because they can couple valuation data across places.
  • Narrowed routeBoundary convexity aloneDifferent binary cusps provide balancing rays, so a proof must also use nonlinear cycle equations, arithmetic quotients, or classification. A combined logarithmic and ordinary formal-immersion argument remains a candidate after the cusp-difference subgroup is controlled.
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 moving-cusp adelic rigidity for bounded-degree exact quadratic cycles outside explicitly classified special loci.

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.

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Prepared starting pointProve moving-cusp adelic rigidity for bounded-degree exact quadratic cycles outside explicitly classified special loci.

Morton–Silverman Uniform Boundedness Conjecture · ready to start

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Research contextPrepared context for any AI agent

For fixed dimension, map degree, and number-field degree, should one bound control every rational preperiodic point of every such dynamical system? The conjecture remains open, including the exact-period obstruction for quadratic polynomials.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references3 cited works · next context review by Nov 15, 2026

The mathematical context was checked on Aug 15, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Rational periodic points of rational functionsoriginal source · Patrick Morton, Joseph H. Silverman · International Mathematics Research Notices · 1994 · DOI 10.1155/S1073792894000127 · accessed Aug 15, 2026
  2. 2
    Current trends and open problems in arithmetic dynamicssurvey or monograph · Robert Benedetto, Patrick Ingram, Rafe Jones, Michelle Manes, Joseph H. Silverman, Thomas J. Tucker · Bulletin of the American Mathematical Society · 2019 · DOI 10.1090/bull/1665 · MR MR4007163 · accessed Aug 15, 2026
  3. 3
    Uniform bounds on periodic points of polynomials with good reductionpeer reviewed result · Isaac Rajagopal, Robin Zhang · Monatshefte für Mathematik (Springer Nature) · 2026-07-22 · ARXIV 2510.26119 · DOI 10.1007/s00605-026-02196-0 · accessed Aug 15, 2026

Important qualifications

  • The review focuses on the full conjecture, a major survey, and one recent special-class theorem; it is not an exhaustive arithmetic-dynamics bibliography.
  • The recent good-reduction theorem treats periodic points for a restricted polynomial class, not all preperiodic points of arbitrary projective morphisms.
  • The bounded review did not establish an end-to-end formalization or a checked general proof certificate.

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