Polynomial algebra, derivatives, and algebraic geometry

Casas–Alvero Conjecture

Collaboration beta

If a polynomial shares a zero with every one of its derivatives, must all of its zeros coincide?

\gcd (f, f^{(j)}) \neq 1 (1 \leq j < N) \Rightarrow f (z) = c (z - a)^{N}

Known results and sources

Casas–Alvero asks whether sharing a zero with every derivative forces the exact form f(z)=c(z-a)^N with c nonzero.

Research problem

Exact mathematical statement

Let $f$ be a complex polynomial of degree $N$ . Suppose $f$ shares a root with each of

f',f'',…,f^{(N-1)}.

The Casas–Alvero Conjecture asserts that there exist $a,c\in\mathbb C$ such that

f(z)=c(z-a)^N.

Problem infographic

Problem at a glance

A problem-first diagram showing a degree-N polynomial, its successive derivatives, a shared root with each derivative, and the exact question whether the polynomial is c times z minus a to the N. — The hypothesis gives a shared root separately for every derivative; the conjecture says this forces one root of multiplicity N.

Current mathematical picture

Where work on Casas–Alvero Conjecture stands

Recent proof claim under review

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

Useful failureBounded degree-20 certificate exhaustion as an all-degree proof

The source explicitly says that no bounded degree-20 computation alone proves the full conjecture. Degree-20 certificate work remains valuable for its exact bounded frontier, while an independent degree-uniform terminal-support theorem must be developed separately.

Route status · Narrowed route

Main reductionAffine mean-root constraint

The A1 and A2 systems must include the exact affine mean-root equation before deeper Hensel expansion.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeIndependently audit the entire A3 certificate-backed elimination from recurrence through component synthesis.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 We removed a duplicate or outdated task or route step.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

Casas–Alvero Conjecture in numbers

3kretained lines of mathematical investigation2,986 in the current working snapshot

Argument development: 2,374 · 80%
Explored or eliminated routes: 216 · 7%
Computational analysis: 170 · 6%
Open obligations: 141 · 5%
Definitions and setup: 85 · 3%

7selected mapped statements2routes 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 byEduardo Casas-AlveroStatus checkedAug 15, 2026

Source recordedAug 15, 2026

Recommended next task

Independently audit the entire A3 certificate-backed elimination from recurrence through component synthesis.

Suggested move: Reconstruct the Appell recurrence and signed base-19 extraction, rerun coefficient-field orbits and saturation, inspect repeated-root arguments, and regenerate the synthesis without the summary JSON.

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

2 recorded

Narrowed routeBounded degree-20 certificate exhaustion as an all-degree proof

Route status · Narrowed route

Narrowed routeConstructing A1 and A2 by modifying reduced A3 equations

The source requires rebuilding A1 and A2 from the integer recurrence rather than modifying reduced A3 equations. Fresh integer-system construction with complete signed digits, bifiltration, residual coefficient fields, and exact mean-root constraints remains viable.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Independently audit the entire A3 certificate-backed elimination from recurrence through component synthesis.Suggested move: Reconstruct the Appell recurrence and signed base-19 extraction, rerun coefficient-field orbits and saturation, inspect repeated-root arguments, and regenerate the synthesis without the summary JSON.

Ready to work on

Eliminate or realize the two live equality supports A1 and A2 using exact algebra.Suggested move: Build both systems anew, adjoin the affine mean-root equation, compute complete finite algebras over every exact coefficient field, and serialize canonical certificates or all surviving components.

Ready to work on

Prove a degree-independent theorem excluding terminal supports at arbitrary depth and degree.Suggested move: Develop one of the source-identified uniform mechanisms: selected-value collision, saturated terminal cluster descent, or degree-independent projective height control.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 15, 2026

Current statusRecent proof claim under review

A March 2026 major revision of arXiv:2501.09272 claims a proof in all characteristic-zero degrees. This collection did not independently verify that recent claim, and the supplied governing source explicitly remains on an open, certificate-audit frontier. The conservative status is therefore open with an unverified claim, not solved.

[3][2][1]

External progress

What the literature has established

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

2026
PreprintA major revised preprint claims a proof for all characteristic-zero degrees; independent validation is not supplied here.[3]
2025
PreprintA finiteness result bounds the dimension of the associated varieties and proves arithmetic finiteness statements toward the conjecture.[2]
2007
Peer reviewedThe conjecture is proved for degrees that are prime powers or twice prime powers; the same paper shows characteristic zero is essential.[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

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 questions2 narrowed routes

Statements by mathematical role7 selected mapped statements

theorem candidate1 of 71
reduction2 of 72
lemma4 of 74

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 statementMust all roots coincide when a polynomial shares a root with every derivative?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementExact derivative-sharing statementintermediate
retained route statementAudited depth boundintermediate
retained route statementA3 internal certificateintermediate
retained route statementAffine mean-root constraintintermediate
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 failureBounded degree-20 certificate exhaustion as an all-degree proofreported failure
Useful failureConstructing A1 and A2 by modifying reduced A3 equationsreported failure
Research targetIndependently audit the entire A3 certificate-backed elimination from recurrence through component synthesis.open
Research targetEliminate or realize the two live equality supports A1 and A2 using exact algebra.open
Research targetProve a degree-independent theorem excluding terminal supports at arbitrary depth and degree.open
Research targetA1 and A2 frontiersuperseded
Research targetAll-depth theoremsuperseded
Narrowed routeBounded degree-20 certificate exhaustion as an all-degree proofThe source explicitly says that no bounded degree-20 computation alone proves the full conjecture. Degree-20 certificate work remains valuable for its exact bounded frontier, while an independent degree-uniform terminal-support theorem must be developed separately.
Narrowed routeConstructing A1 and A2 by modifying reduced A3 equationsThe source requires rebuilding A1 and A2 from the integer recurrence rather than modifying reduced A3 equations. Fresh integer-system construction with complete signed digits, bifiltration, residual coefficient fields, and exact mean-root constraints remains 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 bridgeIndependently audit the entire A3 certificate-backed elimination from recurrence through component synthesis.

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.

Continue the mathematics

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Prepared starting pointIndependently audit the entire A3 certificate-backed elimination from recurrence through component synthesis.

Casas–Alvero Conjecture · 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

If a polynomial shares a zero with every one of its derivatives, must all of its zeros coincide?

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

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Proof attempt or partial resultSupporting notes or data

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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
The Casas-Alvero conjecture for infinitely many degreespeer reviewed result · Hans-Christian Graf von Bothmer, Oliver Labs, Josef Schicho, Christiaan van de Woestijne · Journal of Algebra · 2007-10-01 · ARXIV math/0605090 · DOI 10.1016/j.jalgebra.2007.06.017 · accessed Aug 15, 2026
2
A finiteness result towards the Casas-Alvero Conjecturepreprint · Soham Ghosh · arXiv · 2024-02-28; revised 2025-01-14 · ARXIV 2402.18717 · accessed Aug 15, 2026
3
Proof of the Casas-Alvero conjecturepreprint · Soham Ghosh · arXiv · 2025-01-16; major revision 2026-03-21 · ARXIV 2501.09272 · accessed Aug 15, 2026

Important qualifications

The recent universal proof claim is an arXiv preprint with major revisions and was not independently reviewed here.
No maintained authoritative problem registry resolving the post-claim status was located in this bounded collection.
No submitted-packet URL was fetched and no source-package attachment or certificate was executed or rendered.

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Casas–Alvero Conjecture

Exact mathematical statement

Problem at a glance

Where work on Casas–Alvero Conjecture stands

Casas–Alvero Conjecture in numbers

Independently audit the entire A3 certificate-backed elimination from recurrence through component synthesis.

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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