Algebraic geometry · field theory · resolvent degree · finite groups

Hilbert’s 13th Problem — Algebraic Form

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Can the roots of the general seventh-degree polynomial be built in stages using algebraic functions of only two variables? The known upper bound is three variables; proving that two never suffice remains open.

{RD}_{C} (7) = 3

Hilbert's thirteenth problem

Known results and sources

Hilbert's algebraic thirteenth problem asks whether the general seventh-degree polynomial truly requires algebraic functions of three variables.

Research problem

Exact mathematical statement

Let $\operatorname{RD}_{\mathbf C}(7)$ be the least number of algebraically independent variables needed at each stage of a tower solving the general degree-seven polynomial over $\mathbf C$ . Hilbert's algebraic thirteenth problem asks whether

\operatorname{RD}_{\mathbf C}(7)=3.

The classical upper bound is $\operatorname{RD}_{\mathbf C}(7)\le 3$ , so the unresolved direction is $\operatorname{RD}_{\mathbf C}(7)>2$ . The source passes to the oriented generic septimic and targets $\operatorname{RD}_{\mathbf C}(A_7)>2$ .

Problem infographic

Problem at a glance

A landscape explainer follows a general degree-seven polynomial through towers of two-variable algebraic sheets and contrasts the known three-variable upper bound with the open lower bound. — The known upper bound is RD_C(7) at most three; the open question is whether any tower built from at most two variables per stage can solve the general septimic.

Current mathematical picture

Where work on Hilbert’s 13th Problem — Algebraic Form stands

Open problem

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

Useful failureBridge degree, web classification, or coefficient counting as a standalone obstruction

The source records that a degree-2520 diagonal bridge can itself split through an explicit level-two tower and that one transverse web configuration is essentially equivalent to the original problem. Birational, cohomological, cycle-theoretic, or specialization invariants of primitive skew-surface correspondences may still provide an exclusion.

Route status · Narrowed route

Main reductionCurrent reduction

Use a four-dimensional versal A7-torsor, normalize a hypothetical level-two tower, identify a terminal A7-surface torsor, and reduce the transverse case to a primitive finite closed point or codimension-one/two correspondence.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve the Primitive Skew-Surface Exclusion Theorem for both surviving intersection dimensions.Task status · Ready to work on

Research-record correctionResearch-record correction

We 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.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Hilbert’s 13th Problem — Algebraic Form in numbers

1.2kretained lines of mathematical investigation1,180 in the current working snapshot

Argument development: 1,000 · 85%
Explored or eliminated routes: 33 · 3%
Computational analysis: 15 · 1%
Open obligations: 41 · 3%
Definitions and setup: 91 · 8%

8selected 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.

Proposed byDavid HilbertProposed1900Status checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Prove the Primitive Skew-Surface Exclusion Theorem for both surviving intersection dimensions.

Suggested move: Choose one curve-skew or fully skew stratum and identify an invariant of primitive closed points that is preserved through every level-two stage but fails for the generic A7 torsor.

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

12 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

1 recorded

Narrowed routeBridge degree, web classification, or coefficient counting as a standalone obstruction

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove the Primitive Skew-Surface Exclusion Theorem for both surviving intersection dimensions.Suggested move: Choose one curve-skew or fully skew stratum and identify an invariant of primitive closed points that is preserved through every level-two stage but fails for the generic A7 torsor.

Ready to work on

Audit the primitive closed-point and Hilbert or Chow reformulation.Suggested move: Track finite-flat closure, marked A7 actions, residue-field primitivity, and the outer-automorphism caveat through the birational translation.

Ready to work on

Verify every external and provisional input on the critical dependency path.Suggested move: Check the exact essential-dimension citation, versality equivalences, component bookkeeping, and algebraic-constant descent before treating the reduction as independently established.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen problem

Current peer-reviewed sources state that Hilbert's characteristic-zero septimic conjecture RD_C(7)=3 remains open. The upper bound RD_C(7)<=3 is known, while even RD_C(n)>1 is not known for any general polynomial degree n>=6.

[3][4]

External progress

What the literature has established

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

2026
Peer reviewedEdens and Reichstein explicitly recorded the characteristic-zero sextic, septimic, and octic conjectures as open while studying their failure over some positive-characteristic fields.[4]
2020
Peer reviewedFarb and Wolfson developed resolvent degree systematically and related Hilbert's problem to geometric and finite-group questions.[1]
2010
Peer reviewedDuncan proved essential-dimension results for A7 and S7, including the ed_C(A7)=4 input used by the private packet.[2]
1900
Historical sourceHilbert posed the thirteenth problem; the modern algebraic form asks about the variable complexity of solving the general degree-seven polynomial.[1]

Mathematical neighborhood

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 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

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.

3 standing statements5 proposed statements3 open questions1 narrowed routes

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
reduction3 of 83
lemma2 of 82
equivalence2 of 82

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included

retained route statementDoes the general septimic require three-variable algebraic functions?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementExact resolvent-degree questionintermediate
retained route statementLevel-two towerintermediate
retained route statementVersal A7 modelintermediate
retained route statementSurface-compositum reductionintermediate
retained route statementTwo primitive correspondence typesintermediate
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
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 failureBridge degree, web classification, or coefficient counting as a standalone obstructionreported failure
Research targetProve the Primitive Skew-Surface Exclusion Theorem for both surviving intersection dimensions.open
Research targetAudit the primitive closed-point and Hilbert or Chow reformulation.open
Research targetVerify every external and provisional input on the critical dependency path.open
Research targetPrimitive skew-surface exclusionsuperseded
Narrowed routeBridge degree, web classification, or coefficient counting as a standalone obstructionThe source records that a degree-2520 diagonal bridge can itself split through an explicit level-two tower and that one transverse web configuration is essentially equivalent to the original problem. Birational, cohomological, cycle-theoretic, or specialization invariants of primitive skew-surface correspondences may still provide an exclusion.

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 Primitive Skew-Surface Exclusion Theorem for both surviving intersection dimensions.

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 pointProve the Primitive Skew-Surface Exclusion Theorem for both surviving intersection dimensions.

Hilbert’s 13th Problem — Algebraic Form · ready to start

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Current routes and known obstacles
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Sources and references4 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
Resolvent degree, Hilbert’s 13th Problem and geometrypeer reviewed result · Benson Farb, Jesse Wolfson · L’Enseignement Mathématique 65 · 2020 · ARXIV 1803.04063 · DOI 10.4171/LEM/65-3/4-2 · accessed Aug 14, 2026
2
Essential dimension of A7 and S7peer reviewed result · Alexander Duncan · Mathematical Research Letters · 2010 · ARXIV 0908.3220 · accessed Aug 14, 2026
3
Hilbert's 13th problem for algebraic groupspeer reviewed result · Zinovy Reichstein · L’Enseignement Mathématique 71 · 2025 · ARXIV 2204.13202 · DOI 10.4171/LEM/1075 · accessed Aug 14, 2026
4
Hilbert’s 13th problem in prime characteristicpeer reviewed result · Oakley Edens, Zinovy B. Reichstein · Documenta Mathematica 31 · 2026 · DOI 10.4171/DM/984 · accessed Aug 14, 2026

Important qualifications

Primary peer-reviewed resolvent-degree sources, Duncan's A7 essential-dimension result, and a 2026 paper explicitly stating current characteristic-zero status.
The source material was not treated as external authority. No packet attachment or submitted URL was fetched.
The search is scoped to identity, current status, the standard RD framework, and the external essential-dimension input; it does not verify the current work's internal reduction.

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Hilbert’s 13th Problem — Algebraic Form

Exact mathematical statement

Problem at a glance

Where work on Hilbert’s 13th Problem — Algebraic Form stands

Hilbert’s 13th Problem — Algebraic Form in numbers

Prove the Primitive Skew-Surface Exclusion Theorem for both surviving intersection dimensions.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formalization opportunities

What changed in the research record

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

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