arXiv:2509.24982v2

Bialgebraic varieties of the Gamma function

Sebastian Eterović, Adele Padgett, Roy Zhao

math.CVmath.NT33B1511J9139A45

Paper metadata

Original submission: September 29, 2025
Latest revision: August 11, 2026 · version 2
Claimed MSC: 33B15, 11J91, 39A45
arXiv categories: math.CV, math.NT
Paper license: CC BY-NC-SA 4.0 ↗
Comments: 17 pages. New version includes a second main theorem

Abstract

We characterize the bialgebraic varieties of the $Γ$ function, that is, if $V,W\subseteq\mathbb{C}^n$ are irreducible affine algebraic variety which satisfy $\dim V =\dim W$ and $Γ(V)\subseteq W$ , then the equations defining $V$ (and hence also $W$ ) either give an equality between coordinates, or set some coordinates to be constant. We also classify the case where $V$ and $W$ have the same dimension as varieties over the field of algebraic functions.

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Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 15, 2026

01Statements2 reported findingsContains wrong statements

Both displayed main biconditionals are false as written. Theorem 1.1 permits an arbitrary containing variety $W$ , so its converse fails when $W$ is enlarged. Theorem 1.2 has the same defect, and its requirement $|B|\geq2$ also excludes the elementary case in which all coordinates vary independently. The intended minimal-closure formulations may be repairable, but they are not the statements printed in version 2.

Theorem 1.1Incorrect

The converse fails for a nonminimal containing variety $W$

Page 2 · Theorem 1.1 · arXiv:2509.24982v2

The theorem assumes only $\Gamma(V\cap U_\Gamma^n)\subseteq W$ . Take $n=3$ , $V=\{(z,z,z):z\in\mathbb C\},\qquad W=\{(w_1,w_2,w_3):w_1=w_2\}.$ Both varieties are irreducible, $V$ is trivially bialgebraic, and $\Gamma(V\cap U_\Gamma^3)\subseteq W$ . Nevertheless, $\dim V=1$ while $\dim W=2$ . This directly contradicts the claimed implication from trivial bialgebraicity to equality of dimensions. Version 1 avoided this counterexample by defining $W$ to be the Zariski closure of the image. Repair classification: replace the arbitrary $W$ by $\overline{\Gamma(V\cap U_\Gamma^n)}^{\mathrm{Zar}}$ , or retain only the proved direction $\dim V=\dim W\Rightarrow V$ trivially bialgebraic.

Full paper, version 2 ↗

Theorem 1.2Incorrect

The asserted classification fails in both directions as printed

Page 2 · Theorem 1.2 · arXiv:2509.24982v2

First take $n=1$ , $V=\mathbb C$ , and $W=\mathbb C^2$ . Then $W$ contains the graph of $\Gamma$ and $\dim W=2=2\dim V$ , but no partition of $\{1\}$ can satisfy condition (c), which requires $|B|\geq2$ . Thus the forward implication is false; the intended condition must allow $B=\varnothing$ . Conversely, take $n=2$ , $V=\{(z,z):z\in\mathbb C\},\qquad W=\{(x_1,x_2,y_1,y_2):x_1=x_2\}.$ With $A=C=\varnothing$ and $B=\{1,2\}$ , the stated structural conditions hold, and $W$ contains the graph over $V$ , but $\dim W=3\neq2=2\dim V$ . Hence the reverse implication fails because $W$ may again be enlarged. Repair classification: allow $B=\varnothing$ and require $W$ to be the Zariski closure of the graph over $V$ (or formulate only a one-way result for arbitrary $W$ ).

Full paper, version 2 ↗

02Proofs7 reported findingsContains incorrect or incomplete proofs

The proofs do not establish the printed main theorems. Besides the counterexamples to the statements themselves, the final inductions use generic-fiber conclusions at points of the Gamma graph without proving that those points meet the generic locus, and the proof of Theorem 1.2 invokes a higher-dimensional transcendence bound without the needed argument. Earlier curve arguments also apply Proposition 5.1 at a decay scale not supplied by the preceding estimates. Several additional displayed errors are local and are classified as typos only.

Proof of Theorem 1.1Incomplete as written

The generic-fiber point used in the induction is not justified

Pages 15–16 · first two paragraphs of the proof of Theorem 1.1 · arXiv:2509.24982v2

From $\dim \pi_S(W)=d$ , the fiber-dimension theorem gives zero-dimensional fibers above a dense open subset of $\pi_S(W)$ . The proof then chooses $c\in U$ and assumes that $\Gamma(c)$ belongs to that open subset. This does not follow for the arbitrary containing variety $W$ in the theorem: the set $\Gamma(U)$ may lie wholly in the exceptional locus. The next sentence similarly upgrades a generic statement to every $c\in U$ . If $W$ is instead the Zariski closure of $\Gamma(V)$ , density can potentially supply the missing intersection, but that minimality is absent from the printed hypothesis and the necessary argument is not given. The assertion that $\Gamma$ is locally injective everywhere is also literally false at critical points; only a suitable generic finite-fiber argument is available. Repair classification: no repair is supplied in the paper.

Full paper, version 2 ↗

Proof of Theorem 1.2Incomplete as written

The dimension estimate and generic-locus step need additional arguments

Page 16 · proof of Theorem 1.2 · arXiv:2509.24982v2

The proof infers $2(d-1)\leq\dim\psi_S(W)$ solely from $\dim\pi_S(V)=d-1$ and the transcendence of the one-variable Gamma function. One-variable transcendence does not by itself yield this higher-dimensional lower bound; an induction hypothesis or a functional-transcendence result must be stated and applied. It then declares the set of $c$ for which $W_{(c,\Gamma(c))}$ has generic dimension to be Zariski dense in $\pi_S(V)$ by the fiber-dimension theorem. That theorem gives a dense open subset in the base $\psi_S(W)$ , but it does not ensure that the Gamma graph meets it densely unless an additional minimal-closure argument is made. Finally, the proof presents only the equality-to-structure direction and does not prove the converse for the printed arbitrary $W$ . Repair classification: no complete repair is supplied.

Full paper, version 2 ↗

Claim 6.2 and Proposition 6.4Incomplete as written

The available factorial estimate is applied at the wrong input scale

Pages 14–15 · Claim 6.2 and proof of Proposition 6.4 · arXiv:2509.24982v2

Proposition 5.1 requires an estimate $|f(n)-m_n|<Ce^{-n}$ in the input index $n$ . Lemma 4.4 and Corollary 4.6 instead give a factorially small error in the index $\ell_n$ of the nearby negative integer, of order $\operatorname{poly}(\ell_n)/\Gamma(\ell_n)$ . For a nonconstant algebraic branch one may have $\ell_n\asymp n^q$ with $0<q<1$ , so this estimate need not be $O(e^{-n})$ . In Claim 6.2 the displayed replacement of $\Gamma(\ell_n)$ by $\Gamma(n)$ is not a consequence of Lemma 4.4; Proposition 6.4 makes the same application without a quantitative comparison at all. A plausible repair is to prove a strengthened version of Proposition 5.1 for errors smaller than every power of $n$ , then combine it with the Puiseux growth $\ell_n\asymp n^q$ . That strengthening is not stated or proved here. Repair classification: substantive omitted step; the intended conclusion is not disproved.

Full paper, version 2 ↗

Theorem 1.2(b)Typo

The constant-coordinate projection is mislabelled

Page 2 · Theorem 1.2(b) · arXiv:2509.24982v2

Condition (b) writes $\pi_A:V\to\mathbb C^C$ and says that it maps to coordinates indexed by $A$ . The logic of the partition shows that this must be $\pi_C:V\to\mathbb C^C$ , mapping to coordinates indexed by $C$ . This is a uniquely determined notation correction and does not independently affect the paper's overall correctness status.

Full paper, version 2 ↗

Claim 6.2Typo

The fiber constant is inverted in the invocation of Lemma 4.4

Page 14 · proof of Claim 6.2 · arXiv:2509.24982v2

At a boundary point $(0,c_j)$ of the inverted curve, the displayed relation is $1/\Gamma(\alpha(-n))=c_j$ , hence $\Gamma(\alpha(-n))=1/c_j$ . Lemma 4.4 must therefore be invoked with $1/c_j$ , not $c_j$ . Its local constant becomes proportional to $|c_j|$ , so one should use $\max_{c_j\neq0}|c_j|$ rather than $1/\min_j|c_j|$ ; the printed minimum may also vanish when some $c_j=0$ . These constants can be corrected mechanically, although the separate input-scale gap reported above remains.

Full paper, version 2 ↗

Proposition 5.1Typo

A basepoint is dropped from the last finite-difference display

Page 13 · proof of Proposition 5.1 · arXiv:2509.24982v2

After correctly writing the terms as $(f-q)(r+\ell)$ , the next display changes them to $(f-q)(\ell)$ . The same recurrence works with $(f-q)(r+\ell)$ throughout. This is a local index mismatch with a uniquely determined correction and does not affect the overall status by itself.

Full paper, version 2 ↗

Corollary 3.3Typo

The leading constant in the limit computation is missing an exponent

Page 6 · proof of Corollary 3.3 · arXiv:2509.24982v2

For nonzero integer $a$ , the leading term of the multiplier $m_a(az+b)$ is $a^a z^a$ . The displayed limit therefore yields $a^{a r_2}=1$ , rather than $a^{r_2}=1$ . Because $a\in\mathbb Z\setminus\{0\}$ and $r_2\neq0$ , either relation gives the required conclusion $|a|=1$ . This is a harmless omitted exponent and does not alter the argument's conclusion.

Full paper, version 2 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2509.24982v2
Authors listed: Sebastian Eterović, Adele Padgett, Roy Zhao
Audit date: August 15, 2026

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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. 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For each proof or proof segment, use the narrowest supported outcome: - Correct and complete: every substantive step is verifiable and no case or crucial nontrivial step is missing. - Incorrect as written: a displayed calculation, inference, lemma, or cited application is formally false or inapplicable. - Incomplete as written: a necessary substantive step or case is missing, even though no explicit contradiction has been proved. - Not able to verify: the available paper and sources are insufficient to decide whether the step is valid. Do not flag omitted routine details, compressed arguments, or “analogously” steps when the omitted reasoning is recoverable and correct. For every defect, identify its smallest exact location, explain its downstream dependency, and classify any proposed repair as one of: - Verified repair: supply enough reasoning to verify that the repair works throughout every affected use. - Plausible repair only: describe the possible repair but do not treat it as established. - No repair supplied. Assign the overall proofs status by the following priority: 1. Contains incorrect or incomplete proofs, if at least one proof is Incorrect as written or Incomplete as written. 2. Contains unverified proofs, if there is no proved incorrectness or incompleteness but at least one material proof is Not able to verify. 3. Correct, if all audited proofs are Correct and complete, apart from any typographical or minor formal findings allowed by Part 4. PART 3 — NOVELTY OF THE MAIN STATEMENTS This is a one-sided, evidence-only check. Evaluate only central results. Report a result only when a verifiable primary source, publicly available before the earliest public version of the paper that contains the result, proves the same statement up to a change of notation, or when the result follows directly and immediately from such an earlier theorem. For each non-novelty finding, provide the earlier source, its public date, the exact theorem or proposition, and an explicit hypothesis-and-conclusion mapping. A later source cannot establish non-novelty. Do not flag a result merely because an easier proof exists or because it follows through a nontrivial argument. Do not flag a result that the paper itself explicitly presents as known, a survey item, or an alternative proof. Mention only statements that meet this non-novelty threshold. Say nothing about statements for which no such evidence was found: do not certify novelty, list unsuccessful searches, or add statement-level “no prior source found” entries. If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. Do not report a gap, unsupported statement, minor formal correction, or recommendation to add an explanatory sentence, and do not mention the suppressed item in the report. This suppression does not apply to a genuine typographical error as defined below, because a typo is a literal notation defect that still requires a textual correction. For example, if

P(delta)

is proved for every sufficiently small

delta>0

and

P

is immediately monotone in the required direction, then the claim that

P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

t=0

remains a formal scope mismatch unless the endpoint case was separately established before the application. 7. Apply this mandatory gate before recording any adverse, unverified, typographical, or minor formal finding: (a) identify the exact assertion or inference alleged to be defective; (b) reconstruct the strongest conclusion following from the preceding argument after all admissible immediate formal consequences are added; (c) identify the specific defect that remains — a formally demonstrated falsehood, a concrete nontrivial unresolved proof obligation, or a genuine domain, hypothesis, quantifier, or notation mismatch requiring a textual correction; and (d) state the evidence and minimal repair. If no specific defect remains, or if the only possible addition would be explanatory, suppress the finding entirely. Use Not able to verify only when a precise, nontrivial unresolved obligation can be identified. 8. A typographical error is an incorrect symbol, index, exponent, sign, variable name, or notation mismatch whose unique intended correction is mechanically determined from the surrounding text and does not alter the mathematical argument. Report it in yellow as Typo, with the printed text, correction, exact location, and reason it is harmless. It does not lower an overall status. 9. A minor formal correction is a boundary, endpoint, or displayed parameter-range defect that can be repaired consistently in every affected statement by a local change, without a new argument, without invalidating any related result or application, and without creating another substantive defect. Report it in yellow as Minor formal correction, with the same details. It does not lower an overall status. 10. Do not use either minor category if more than one materially different repair is possible; if the repair changes the substance or scope of a central result; if an excluded case is used later; if the correction fails to propagate consistently; or if a new mathematical idea is required. In such cases use the strongest substantive classification actually supported by evidence. 11. Write every mathematical expression as valid, compilable LaTeX. Use

...

for inline mathematics and

...

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Bialgebraic varieties of the Gamma function

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The converse fails for a nonminimal containing variety WWW

The asserted classification fails in both directions as printed

The generic-fiber point used in the induction is not justified

The dimension estimate and generic-locus step need additional arguments

The available factorial estimate is applied at the wrong input scale

The constant-coordinate projection is mislabelled

The fiber constant is inverted in the invocation of Lemma 4.4

A basepoint is dropped from the last finite-difference display

The leading constant in the limit computation is missing an exponent

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The converse fails for a nonminimal containing variety $W$