--- title: "Smooth-Point Certificates for Polydegree Containments: A Jacobian Interpretation of the Lewis-Perry-Straub Determinant" date: 2026-08-09 version: "0.4.1-candidate" doi: 10.5281/zenodo.21864574 pdf: 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 repository: https://github.com/ipitchford/smooth-point-certificates-polydegree-containments archive: https://zenodo.org/records/21864574 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # Smooth-Point Certificates for Polydegree Containments: A Jacobian Interpretation of the Lewis-Perry-Straub Determinant ## Plain-English summary Imagine sorting polynomial transformations of the plane by the sequence of degrees used to build them. The Polydegree Conjecture asks when one such family can occur as a limiting case of another. Earlier work turned many of these geometric questions into an explicit search for numbers that make several polynomials zero while keeping another polynomial and a determinant non-zero. This candidate explains what that apparently extra determinant means. It is, up to sign, the **Jacobian determinant** of the coefficient polynomials already in the problem. A Jacobian detects whether equations meet cleanly at a point. The old algebraic test can therefore be read as a much more familiar geometric request: find one smooth point of an intersection that does not lie on the next forbidden hypersurface. That change of viewpoint also makes the computer certificate smaller. Instead of evaluating a separate determinant formula, a search over a finite field can look for a simple common zero and check that the next coefficient is non-zero. Hensel lifting then carries the point to characteristic zero. The paper proves this reformulation and several supporting theorems. It also reports exact calculations in the first substantial three-variable family. Those calculations succeed for every index from 2 through 20, but they are finite evidence, not a proof for every index. > **Candidate status:** anonymous · unrefereed · exact producer replay passed · > no independent reproduction, proof-assistant formalisation, external > specialist review, or editorial peer review. ## The central claim or finding Lewis, Perry and Straub define coefficient polynomials $g_{n,e}$, matrix entries $\alpha_{i,j,e}$, and an auxiliary determinant $a_{d,e}$. The key identity is $$ \alpha_{i,j,e}=-\frac{\partial g_{j+1,e}}{\partial x_{i+1}}, \qquad a_{d,e}=(-1)^e\det D(g_{d,e},\ldots,g_{d+e-1,e}). $$ Thus the determinant is not independent machinery. It is the Jacobian of the coefficient map already being studied. Weighted homogeneity then gives an exact integral syzygy. On the partial zero locus, that identity reduces the full determinant test to two local conditions: the next coefficient must be non-zero, and the smaller affine Jacobian minor must be non-zero. In the chart $x_1=1$, this says precisely that the point is smooth and avoids the next coefficient hypersurface. The resulting finite-field criterion is sharper than the legacy modular test. A simple common zero with non-zero next coefficient lifts to the required characteristic-zero point even if the chosen prime divides the scalar that appears in the Euler identity. The residue-field determinant may then vanish, but its value at the exact $p$-adic lift is still non-zero. ## How the result works There are four linked arguments. First, a coefficient reindexing turns every entry of the auxiliary matrix into a partial derivative. Taking determinants gives the Jacobian identity. Second, the coefficient polynomials carry natural weights. Expanding their weighted Euler identities along the first column produces an integral relation between the full determinant, the next coefficient, the affine Jacobian minor, and the equations that already vanish. Because the relation is integral, it can be reduced modulo primes without first dividing by a potentially vanishing integer. Third, multivariate Hensel lifting starts from a simple finite-field zero and produces an exact point over the $p$-adic integers. Localisation, faithful base change, and the weak Nullstellensatz then transfer existence to the complex numbers. The archive retains the case $d=8$, $p=5$ as a regression example: the refactored criterion works although the old demand that the full determinant be a unit modulo $p$ does not. Fourth, a separate geometric bridge works on a smooth complete-intersection curve. If the next equation cuts out a non-empty reduced affine zero divisor and the final coefficient avoids its support, at least one point supplies the required Jacobian certificate. The closed-point statement is made over a perfect ground field before base change, which avoids assigning a spurious degree to a point over an algebraically closed field. For $e=3$, exact rational and Singular calculations find $$ \dim_{\mathbf Q}\frac{\mathbf Q[X,Y]}{(G_d,G_{d+1})} =\left\lfloor\frac{d(d+1)}6\right\rfloor $$ for $2\le d\le20$, with the Jacobian and $G_{d+2}$ invertible in every coordinate algebra. A separate hypergeometric argument proves that every individual one-variable boundary factor has simple positive roots, hence is squarefree, for all indices. ## What is classical, and what is offered here The underlying Polydegree containment criterion, its coefficient polynomials, and the algorithmic programme belong to Lewis, Perry and Straub. Perry's thesis developed the computational direction for the case later indexed as $e=3$. Lewis--Perry--Straub also record an existing infinite family in that column, attributed to Edo. None of those results is claimed here as new. The candidate contribution is the derivative--Jacobian interpretation, the integral weighted-Euler reduction, the smaller Hensel certificate, the curve-theoretic bridge, and the uniform squarefreeness theorem for each individual boundary factor. The exact affine and adjacent-coprimality sweeps are supporting finite evidence. A bounded literature search did not locate the same Jacobian packaging, but that is not a priority certificate. Original-author feedback, specialist database searches, and external review could change the historical assessment. ## Evidence and audit trail The immutable package contains a 17-page paper, accessible Markdown and LaTeX, the bibliography and source records, exact evidence, review records, and a SHA-256 manifest covering 142 durable files. The replacement standard-library verifier performs 10 symbolic gates and checks all 51 frozen witnesses. It passes under ordinary Python and `python -O`. Sixteen mutation or negative controls and one positive bad-characteristic regression also pass in both modes. Exact geometry receipts cover the affine family through $d=20$, boundary arithmetic through $d=200$, and the wider ranges stated in the package. Citation review covered all nine bibliography records and all 14 in-text contexts. An originality screen sampled 40 of 132 body paragraphs across every major section and found no close or verbatim match in the checked corpus. These are checks made inside the producing workflow. They can expose broken formulas, corrupted files, stale builds, and fail-open code paths. They do not turn the paper into an independently reproduced or peer-reviewed result. One upstream verifier defect is deliberately preserved in the record: a proof-critical integrality guard used Python `assert`, which disappears under optimized execution. The current witness data still pass, and the replacement verifier uses explicit exceptions. This is a robustness risk in the old checker, not evidence that a current certificate is false. ## What the result does not establish - It does not prove the full Polydegree Conjecture or the Polydegree Ideal Conjecture. - It does not prove the uniform $e=3$ affine statement for every $d$. - It does not prove uniform coprimality of adjacent boundary factors; that property is checked exactly only through $d=200$. - It does not establish absolute novelty or priority. - It has not been independently reconstructed, formally verified, reviewed by an external specialist, or accepted by a journal. - A public DOI, passing CI run, and matching hashes establish availability and integrity of the released bytes; they do not establish mathematical truth. ## Relationship to earlier work Lewis, Perry and Straub supply the specialisation theorem to which the new smooth-point certificate is applied. Their Theorem 4 is a sufficient containment criterion; this candidate does not claim it is necessary. Perry's dissertation is an antecedent for the $e=3$ computational programme, while Edo and van den Essen provide the surrounding Strong Factorial context. The boundary proof uses two results on finite free convolution by Martínez-Finkelshtein, Morales and Perales. The repaired proof reflects one Jacobi factor and invokes logarithmic-mesh preservation to obtain simple roots; it does not rely on a stronger strict-preservation sentence that is false in the generality printed elsewhere. ## Who should care, and why | Audience | Potential use | Required caution | |---|---|---| | Polynomial-automorphism researchers | Replace an auxiliary determinant condition with a local smoothness problem. | The all-$d$ affine theorem remains open. | | Algebraic geometers | Study explicit complete intersections through Jacobians, divisor support, and weighted identities. | The uniform radicality and disjointness claims are conjectural. | | Arithmetic and computational algebra researchers | Search with a smaller modular certificate and use primes rejected by the legacy determinant-unit test. | Producer replay is not an unaffiliated reimplementation. | | Special-functions researchers | Reuse the Jacobi and finite-free-convolution route to individual boundary squarefreeness. | Adjacent coprimality compares changing families and is still unproved. | | Computer-assisted mathematics researchers | Inspect a package that distinguishes proof, finite computation, controls, and open claims. | Hashes and CI certify bytes and replay, not the theorem itself. | ## How to reproduce the recorded checks Download the ZIP from the immutable GitHub prerelease or Zenodo and verify its `MANIFEST.sha256`. The package `BUILD.md` gives the exact commands. The core portable route is: ```sh python3 -B scripts/verify_manifest.py --root "$PWD" python3 -B -O scripts/verify_manifest.py --root "$PWD" python3 -B scripts/validate_candidate.py --root "$PWD" --publication-ready python3 -B -O scripts/validate_candidate.py --root "$PWD" --publication-ready python3 -B scripts/verify_jacobian_refactor.py \ --witness-data evidence/upstream/polydegree_e3_witnesses_50_100.json python3 -B scripts/test_mutations.py \ --witness-data evidence/upstream/polydegree_e3_witnesses_50_100.json ``` The exact affine and boundary regenerations additionally require SymPy 1.14.0 and Singular 4.4.1. A stronger independent check would reconstruct the identities and geometry from the definitions without consulting the supplied verifier outputs. ## The most valuable next projects 1. Prove or disprove uniform radicality and the predicted length of $(G_d,G_{d+1})$, with both $J_d$ and $G_{d+2}$ invertible. 2. Prove uniform adjacent boundary coprimality, or find the first counterexample across the changing hypergeometric families. 3. Exclude triple common zeros of $G_d,G_{d+1},G_{d+2}$ structurally rather than by extending a finite table. 4. Obtain an unaffiliated proof reconstruction, original-author feedback, and external algebraic-geometry or polynomial-automorphism review. 5. Formalise the derivative identities, Euler syzygy, and Hensel bridge in a proof assistant. ## What is in the public package The public record contains the candidate paper in PDF, LaTeX, text, and accessible Markdown; bilingual abstracts; the verified bibliography; frozen finite-field witnesses; exact affine and boundary receipts; the uniform boundary-squarefreeness proof and referee repair; the portable Jacobian verifier; mutation controls; provenance, claim-status, citation, originality, and assurance records; licensing information; CI configuration; build instructions; and the complete manifest. The explanatory cover, Open Graph card, transcript, audio briefing, and video thumbnail belong to the Evidence Press presentation layer. They are not mathematical evidence. ## Open directions for follow-up research - 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. ## Verification status 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. ## Sources and related work - 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. - Perry, K. A. (2016). Polydegree properties of polynomial automorphisms. Doctoral dissertation, The University of Alabama. - Edo, E., & van den Essen, A. (2014). The Strong Factorial Conjecture. Journal of Algebra, 397, 443-456. - Grochow, J. A. (2025). Multivariate Hensel lifting without assuming as many variables as equations. The American Mathematical Monthly, 132(4), 350-355. - Martinez-Finkelshtein, A., Morales, R., & Perales, D. (2024). Zeros of generalized hypergeometric polynomials via finite free convolution. Applications to multiple orthogonality. arXiv:2404.11479v2. - Martinez-Finkelshtein, A., Morales, R., & Perales, D. (2024). Real roots of hypergeometric polynomials via finite free convolution. arXiv:2309.10970v3.