--- title: "An eight-term bound for degree-twenty Casas–Alvero polynomials" date: 2026-09-24 version: "0.1.0-candidate" doi: 10.5281/zenodo.22943640 pdf: https://github.com/ipitchford/casas-alvero-eight-term/releases/download/v0.1.0-candidate/casas-alvero-eight-term-v0.1.0-candidate.pdf repository: https://github.com/ipitchford/casas-alvero-eight-term archive: https://zenodo.org/records/22943640 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # An eight-term bound for degree-twenty Casas–Alvero polynomials ## Summary A polynomial can share a root with one derivative without anything unusual happening. The Casas–Alvero conjecture asks what happens when it shares a root with **every** proper derivative. In characteristic zero, the conjectured answer is that all its roots must be the same. This candidate studies degree twenty and rules out counterexamples with seven or fewer nonzero terms after a specific centering. Any nontrivial counterexample would need **at least eight terms**, including the leading term. The full degree-twenty conjecture is not settled. ## Summary for specialists Let $f$ be a nontrivial characteristic-zero Casas–Alvero polynomial of degree twenty. Translate the root of $f^{(19)}$ to zero and make $f$ monic. Writing $$f(X)=X^{20}+\sum_{j\in S}c_jX^{20-j},\qquad c_j\ne0,$$ the candidate theorem gives $1+|S|\ge8$. Here $S\subseteq\{2,\ldots,19\}$ consists of deficiency indices, not exponents. An arbitrary translation need not preserve the bound. Published arithmetic restrictions leave fourteen exact seven-term supports. The complete characteristic-seventeen classification leaves nineteen compatible support/seed cases, and every one is excluded. Coefficients may be nonzero while reducing to zero, and arbitrary finite ramification is allowed. ## Technical account The proof combines exact finite calculations with arguments about actual roots. The important step is not merely finding no solutions in a finite field: a hypothetical characteristic-zero solution must be shown to enter one of the checked cases. One reusable argument concerns a separated cluster of $m$ roots. If the cluster contains a common witness for every Hasse derivative of orders one through $m-1$, and the degree-$m$ Casas–Alvero property holds over the residue-field closure, then the roots in that cluster coincide exactly. Otherwise, rescaling their largest internal separation gives a forbidden nontrivial degree-$m$ polynomial. The paper applies this with four roots in residue characteristic seventeen. Other cases use an incompatible second jet or a quadratic obstruction. A unit-Jacobian precision lemma justifies the transfer to ramified candidates. These tools are classical; the proposed contribution is the complete degree-twenty exclusion and its specific local arguments. Uniformity of one coefficient-stratum theorem does not mean uniformity across degrees. For example, the seven-term support $S=\{2,4,10,17,18,19\}$ would allow powers $20,18,16,10,3,2,1$. It has three compatible seed cases before an exact-zero restriction. The proof covers all three; it does not select only coefficients with nonzero residues. The cover graphic shows the term-count boundary schematically, not a numerical root plot. ## Evidence, assurance and limitations The package contains the written proof, all required finite inputs and checkers, and a complete support-to-proof map. The revised primary runner executes sixteen checks in normal and optimized Python, including the previously omitted census of 1,216 row-8 assignments. Its thirty-two executions pass locally. The inherited runner passes twenty-seven checks with optional SymPy installed, or twenty-six with that named omission explicitly recorded. Deliberately corrupted domains, support inventories and arithmetic claims are rejected. These are producer-side checks. Internal editorial roles, reported external review activity, exact hashes and a DOI do not establish formal verification or journal peer review. The latest supplied review's linked audit ZIP was unavailable at intake, so its reported reproduction is not silently promoted to authenticated external assurance. The theorem does not establish sharpness or an eight-term counterexample. ## Relationship to earlier work Castryck, Laterveer and Ounaïes supply the mean-root and missing-index determinant restrictions. De Frutos Marín supplies older sparse-support criteria. Massri's degree-twenty result concerns three recycled roots, which is a different invariant. Marashdeh provides related support reductions. Ghosh's March 2026 preprint claims the full conjecture; this paper does not use it as a premise. An objection to a positive-characteristic auxiliary claim does not itself refute a characteristic-zero conclusion. The bounded comparison found no exact predecessor for the eight-term theorem. Equivalence with some reported systems and an unavailable thesis remain unresolved, so unconditional priority is not claimed. ## Who should care, and why | Audience | Potential use | Required caution | |---|---|---| | Algebraists studying Casas–Alvero | Inspect a complete sparsity-class exclusion | The unrestricted degree-twenty problem remains outside this result | | Computer-algebra researchers | Reproduce the finite identities and support cover | The encoded calculation needs its written mathematical transfer | | Researchers using valuations | Examine the separated-cluster and precision arguments | Retain every witness and residue-field hypothesis | | Formalization researchers | Identify bounded components for certified checking | The current package is not a proof-assistant formalization | ## Why the problem matters A sparsity theorem restricts the shape of any possible counterexample and supplies concrete local obstructions that another argument may reuse. It is useful only if the exclusions cover the entire stated class. Fourteen supports and nineteen cases are bookkeeping counts, not a percentage of progress toward the full conjecture. ## How to inspect or reproduce the recorded checks Extract the versioned archive and use Python 3.10 or later. The primary runner needs only the standard library; the inherited runner also needs a C++ compiler, with SymPy optional for one historical cross-check. ```sh python3 -B replay.py --integrity-only python3 -B replay_new.py --output ../new-replay python3 -B replay.py --output ../inherited-replay python3 -B test_publication.py ``` Read the actual omission list and scope in each receipt. The optional large historical census is not a dependency of the eight-term theorem. ## What is in the evidence package Start with `AI_INDEX.md` and the numbered statements in `PAPER.pdf`. The structural map identifies the main theorem, uniform stratum obstruction, precision lemma and cluster lemma. `research/next-stage/coverage/` routes every support/seed case. `new-results/check_row8_seven_term.py` checks the complete small census. The package also includes source files, certificates, current replay logs, internal editorial records and a revision response. Historical work remains labelled separately from the current candidate. ## What would improve the result next The immediate task is specialist examination of the normalization and ramification arguments, alongside a separately implemented replay of their finite inputs. Extending the theorem would then require new obstructions for denser centered supports; the current eight-term bound gives no guarantee that those cases are tractable. ## Open directions for follow-up research - Obtain unaffiliated specialist examination of the full valuation argument and independent reproduction of its finite dependencies. - Determine which additional degree-twenty strata can be excluded while allowing arbitrary ramification. - Test further applications of the separated-cluster lemma with every derivative-witness hypothesis retained. ## Research process, metrics and reusable methods This is prospective process metadata under the Evidence Press operating model and research-metrics policy. It records the intended handoff, measured scope and claim boundary; it is not evidence that the method accelerated this work. - Work ID: ep-work:casas-alvero-eight-term - Attempt and metric receipts: ep-attempt:casas-alvero-eight-term-publication: published / positive; scope assurance-through-publication; target Internal editorial approval, immutable research assets and canonical Evidence Press availability.; active forecast 150 minutes (100-240); Fermi components Remaining exposition and deterministic readiness: 1 x 25/40/65 minutes low/central/high (Procedural prior; no comparative acceleration claim.); Internal editorial gate: 1 x 25/35/60 minutes low/central/high (Procedural prior; no comparative acceleration claim.); Archive, media and guarded deployment: 1 x 50/75/115 minutes low/central/high (Procedural prior; no comparative acceleration claim.); positive-signal/closure probabilities 0.95/0.85 within 240 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 29/unknown/unknown/2/0/3; cycles positive/negative/inconclusive 0/0/0; falsification gates 4; architectures tested/rejected 1/0; result target-closed; target reached true; forecast error -121 minutes; ratio 0.1933; inside interval false; positive-signal/target-closure Brier scores 0.0025/0.0225; missing telemetry activeHumanMinutes: Human effort not instrumented.; computeMinutes: No separately instrumented total; ordinary validation excluded.; deduplicatedModelTokens: Fork-aware runtime counter unavailable.; uncachedInputTokens: Runtime token/cache accounting unavailable.; appended measurement corrections measurement.reworkMinutes -> metrics.outcome.reworkMinutes: metrics.outcome.reworkMinutes = 3, a rounded estimate within measured root active time through first canonical availability; no separate rework stopwatch. (reason: The first public registration retained a zero initialization for rework, not a separately timed no-rework observation. Preserve that historical field and explicitly distinguish the later rounded allocation estimate in the outcome.). Work ledger: https://evidencepress.org/api/work-ledger.json. Metrics policy: https://evidencepress.org/api/research-metrics-policy.json - Intended aims: science - Artifact roles: research-output, evidence-assessment, communication - Decision object: bound — Eight-term lower bound for centered degree-twenty Casas–Alvero counterexamples. Scope: Characteristic zero, nontrivial polynomial, centering at the nineteenth-derivative root; leading monomial included. - Reusable methods: Structural compression (structural-compression); Adversarial scientific controls (adversarial-controls); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json - Targeted clocks: assurance, publication - Semantic bridge: explicit — Exact support-preserving specialization and integral normalization lead to a complete seed cover. Written root-cluster and precision arguments transfer exact finite obstructions back to characteristic-zero candidates. Remaining risks: Long valuation arguments need external specialist scrutiny.; Finite-field absence alone does not prove characteristic-zero exclusion.; Priority comparisons remain bounded and partly unresolved.. - Human judgement gates: Assess the source-to-claim correspondence and written proof.; Preserve rights, status and priority boundaries.; Publication is authorised; external review is a separate dimension. - Next assurance action: Inspect and independently reproduce the bounded result; explore extensions separately. External review is not a publication prerequisite. - Claim ceiling: Unrefereed candidate. The full degree-twenty conjecture, formal verification, external journal acceptance and unconditional priority are not established. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — Inspectable degree-twenty sparsity restriction in Producer-coordinated mathematical publication; design none; comparator None.; estimand No acceleration or impact effect estimated.; no real-world effect evidence asserted ## Verification status Unrefereed candidate. The full degree-twenty conjecture, formal verification, external journal acceptance and unconditional priority are not established. ## References 1. Castryck, Laterveer and Ounaïes (2014): mean-root simplicity, determinant restrictions and degree-twelve computation. 2. De Frutos Marín (2013): arithmetic and sparse-support criteria. 3. Massri (v6, 2023): degree twenty with three recycled roots, a different invariant. 4. Marashdeh (2026): descent-set and support obstructions. 5. Ghosh (v2, March 2026): unrefereed all-degree proof claim, not a premise of this paper.