{
  "schemaVersion": "1.2",
  "slug": "smooth-point-certificates-polydegree-containments",
  "title": "Smooth-Point Certificates for Polydegree Containments: A Jacobian Interpretation of the Lewis-Perry-Straub Determinant",
  "shortTitle": "Smooth-point certificates for Polydegree containments",
  "url": "https://evidencepress.org/releases/smooth-point-certificates-polydegree-containments/",
  "oneLine": "The Lewis-Perry-Straub auxiliary determinant is a Jacobian: an exact Euler syzygy turns its specialisation test into a smooth-point criterion and yields a smaller Hensel certificate, while the uniform e=3 affine theorem remains open.",
  "abstract": "Lewis, Perry and Straub introduced integral coefficient polynomials and an auxiliary determinant whose complex specialisations imply Polydegree containments for plane polynomial automorphisms. This unrefereed candidate identifies the determinant, up to sign, as the Jacobian of the coefficient map and derives an exact weighted-Euler syzygy. In the chart x1=1, the original non-vanishing condition becomes the existence of a smooth point of the partial coefficient complete intersection outside the next coefficient hypersurface. A simple finite-field common zero with non-zero next coefficient therefore lifts to the required characteristic-zero point; no restriction excluding primes dividing d+e-1 is needed. A separate curve theorem shows how a reduced affine zero divisor supplies the same point. Exact e=3 computations prove that the entire affine intersection is reduced and avoids the next coefficient for 2 <= d <= 20. A terminating-hypergeometric and finite-free-convolution argument proves every individual boundary factor has only simple positive roots, while adjacent coprimality is checked through d=200. The all-d affine conjecture remains open. Internal exact replay, code-path diversity and controls are not represented as independent reproduction, formal verification, specialist review or peer review.",
  "datePublished": "2026-08-09",
  "dateModified": "2026-08-09",
  "version": "0.4.1-candidate",
  "doi": "10.5281/zenodo.21864574",
  "doiUrl": "https://doi.org/10.5281/zenodo.21864574",
  "conceptDoi": "10.5281/zenodo.21864573",
  "pdfUrl": "https://github.com/ipitchford/smooth-point-certificates-polydegree-containments/releases/download/v0.4.1-candidate/smooth-point-certificates-polydegree-containments-v0.4.1-candidate.pdf",
  "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/smooth-point-certificates-polydegree-containments/v0.4.1-candidate/build/main.pdf",
  "zenodoUrl": "https://zenodo.org/records/21864574",
  "repoUrl": "https://github.com/ipitchford/smooth-point-certificates-polydegree-containments",
  "releaseUrl": "https://github.com/ipitchford/smooth-point-certificates-polydegree-containments/releases/tag/v0.4.1-candidate",
  "markdownUrl": "https://evidencepress.org/releases/smooth-point-certificates-polydegree-containments/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/smooth-point-certificates-polydegree-containments/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/smooth-point-certificates-polydegree-containments.mp3",
  "imageUrl": "https://evidencepress.org/assets/og/smooth-point-certificates-polydegree-containments.png",
  "coverArtUrl": "https://evidencepress.org/assets/art/smooth-point-certificates-polydegree-containments.svg",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/smooth-point-certificates-polydegree-containments.mp3",
      "name": "Audio briefing -- smooth-point certificates for Polydegree containments",
      "description": "Plain-English AI-generated voice summary of the central Jacobian interpretation, evidence boundary, and open uniform theorem.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/smooth-point-certificates-polydegree-containments.txt"
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Anonymous, unrefereed candidate. The derivative, determinant, weighted-Euler, Hensel/localisation, curve-bridge and uniform boundary-squarefreeness results have written proofs and internal adversarial checks. Exact finite computations are bounded at d <= 20 for the full affine statement and d <= 200 for adjacent boundary coprimality. The uniform e=3 affine theorem, full Polydegree Conjecture and priority question remain open. The public repository, immutable candidate release and Zenodo record expose byte-identical assets. No unaffiliated party has independently reconstructed the proofs or computations; there is no proof-assistant formalisation, external specialist audit, editorial peer review or venue acceptance. An upstream proof-critical Python assert is retained as a disclosed fail-open robustness defect, with no evidence that a current witness is false."
  },
  "assurance": [
    {
      "dimension": "availability",
      "label": "Availability and archiving",
      "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?",
      "state": "passed",
      "evidenceUrl": "https://zenodo.org/records/21864574",
      "note": "The exact candidate PDF, 142-file ZIP and checksum ledger are public on Zenodo and the immutable GitHub prerelease; downloaded bytes match."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "The manifest, candidate validator, 10/10 symbolic gates, 51/51 witnesses and 17/17 controls pass under ordinary and optimised Python on the immutable tag.",
      "evidenceUrl": "https://github.com/ipitchford/smooth-point-certificates-polydegree-containments/actions/runs/31336017231"
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "No unaffiliated person or group has reported rerunning the immutable public package."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "Different code paths and computer-algebra routes were produced inside one coordinated workflow and are not independent reproduction."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "No theorem has been formalised in a proof assistant."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "No external polynomial-automorphism or algebraic-geometry specialist has reviewed the candidate."
    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
      "state": "not-assessed",
      "note": "No journal or venue has conducted peer review."
    },
    {
      "dimension": "dataEnvironmentReproducibility",
      "label": "Data and environment reproducibility",
      "question": "Are data and computational environment pinned well enough to rebuild?",
      "state": "partial",
      "evidenceUrl": "https://github.com/ipitchford/smooth-point-certificates-polydegree-containments/blob/v0.4.1-candidate/BUILD.md",
      "note": "Exact scripts, receipts, dependency versions, commands and CI are supplied, but no independently recreated container or cross-platform replay is available."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "Automated research, symbolic-computation and language-model tools"
    ],
    "humanRole": "Problem selection, research direction, mediation, supervision, review and publication decision; the scholarly creator is cited as Anonymous.",
    "disclosure": "Automated research, symbolic-computation and language-model tools assisted literature triage, proof exploration, exact verification, adversarial review, drafting and package preparation. They are not authors. Agreement among AI-assisted checks is not independent human review."
  },
  "problem": {
    "name": "Polydegree Conjecture for plane polynomial automorphisms",
    "url": "https://doi.org/10.1016/j.jpaa.2019.04.002"
  },
  "corrections": [],
  "keywords": [
    "plane polynomial automorphisms",
    "Polydegree Conjecture",
    "Jacobian criterion",
    "multivariate Hensel lifting",
    "weighted homogeneous polynomials",
    "complete intersections",
    "computer-assisted mathematics",
    "reproducible research",
    "unrefereed candidate"
  ],
  "keyResults": [
    "The Lewis-Perry-Straub auxiliary determinant is, up to sign, the full coefficient Jacobian determinant.",
    "An exact integral weighted-Euler syzygy reduces the determinant condition on the partial zero locus to non-vanishing of the next coefficient and an affine Jacobian minor.",
    "A simple finite-field common zero with non-zero next coefficient gives the required complex specialisation; the prime need not avoid d+e-1.",
    "A reduced affine zero divisor on the scheme-theoretic smooth complete-intersection curve supplies the same Jacobian point.",
    "For e=3 and 2 <= d <= 20, exact rational and Singular computation shows the full affine intersection is reduced with the predicted length and avoids the next coefficient.",
    "Every e=3 one-variable boundary factor is uniformly squarefree; adjacent-factor coprimality is exactly checked through d=200 and remains open uniformly."
  ],
  "reviews": [],
  "evidencePackage": "A 17-page candidate paper in PDF, LaTeX and accessible Markdown; bilingual abstracts; exact proof record; frozen 51-witness corpus; upstream replay reconciliation; a no-import standard-library Jacobian verifier with 10/10 symbolic gates and 51/51 witness relations under ordinary and optimised Python; 16 negative or mutation controls plus one positive bad-characteristic regression; exact rational and Singular affine computations for 2 <= d <= 20; an exact boundary-factor verifier and repaired referee memo; adjacent boundary arithmetic through d=200; citation, originality, claim-status, provenance and assurance records; and a SHA-256 manifest covering 142 release files. These are producer-side checks, not independent reproduction or external theorem verification.",
  "openProblems": [
    "Prove or disprove that (G_d,G_(d+1)) is radical of the predicted length, with J_d and G_(d+2) units, for every d >= 2.",
    "Prove uniform adjacent boundary coprimality across the changing terminating-hypergeometric families, or locate a counterexample.",
    "Prove or refute triple common-zero exclusion for G_d,G_(d+1),G_(d+2) in the e=3 affine chart.",
    "Obtain original-author feedback, authenticated specialist-database searches and an external priority assessment of the Jacobian packaging.",
    "Independently reconstruct the proof chain and computations, and formalise the algebraic identities and Hensel bridge in a proof assistant."
  ],
  "relatedWorks": [
    {
      "citation": "Lewis, D., Perry, K., & Straub, A. (2019). An algorithmic approach to the Polydegree Conjecture for plane polynomial automorphisms. Journal of Pure and Applied Algebra, 223(12), 5346-5359.",
      "url": "https://doi.org/10.1016/j.jpaa.2019.04.002"
    },
    {
      "citation": "Perry, K. A. (2016). Polydegree properties of polynomial automorphisms. Doctoral dissertation, The University of Alabama.",
      "url": "https://ir.ua.edu/items/f71aa1b3-0dbc-42a2-9f59-3f01eb1a5529"
    },
    {
      "citation": "Edo, E., & van den Essen, A. (2014). The Strong Factorial Conjecture. Journal of Algebra, 397, 443-456.",
      "url": "https://doi.org/10.1016/j.jalgebra.2013.09.011"
    },
    {
      "citation": "Grochow, J. A. (2025). Multivariate Hensel lifting without assuming as many variables as equations. The American Mathematical Monthly, 132(4), 350-355.",
      "url": "https://doi.org/10.1080/00029890.2024.2440297"
    },
    {
      "citation": "Martinez-Finkelshtein, A., Morales, R., & Perales, D. (2024). Zeros of generalized hypergeometric polynomials via finite free convolution. Applications to multiple orthogonality. arXiv:2404.11479v2.",
      "url": "https://arxiv.org/abs/2404.11479v2"
    },
    {
      "citation": "Martinez-Finkelshtein, A., Morales, R., & Perales, D. (2024). Real roots of hypergeometric polynomials via finite free convolution. arXiv:2309.10970v3.",
      "url": "https://arxiv.org/abs/2309.10970v3"
    }
  ]
}