arXiv:2607.09565v1

Explicit height bounds for G-functions and unlikely intersections with lines in tori

Martin Orr

math.NT11G3011G5011J91

Paper metadata

Original submission: July 10, 2026
Claimed MSC: 11G30, 11G50, 11J91
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 38 pages

Abstract

This paper computes explicit constants in Bombieri and André's height bound for points at which there are unexpected "global" polymomial relations between values of G-functions. It applies these bounds for G-functions to obtain explicit height bounds for unlikely intersections with lines in tori, making explicit a weak version of the bounded height theorem of Bombieri, Masser and Zannier and, in some cases, improving an explicit bound of Habegger.

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Audit summary

Audited against arXiv v1

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.

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Detailed mathematical audit

Generated August 15, 2026

01Statements4 reported findingsCorrect

The refined explicit height bound for global relations between values of G-functions and its application to unlikely intersections with lines in algebraic tori are correct. One harmless place-type typo is reported in yellow.

Theorem 4.2 and Corollary 4.5Correct

The explicit G-function height bounds are correct

Pages 16–22 · Theorem 4.2 and Corollaries 4.4–4.5 · arXiv:2607.09565v1

The corrected André inequality is applied to the $\lambda$ monomial values and $\kappa$ independent global relations with the required ordinary-point and linear-independence hypotheses. Lemmas 4.7 and 4.8 transfer the size, radius, and differential-operator data to the monomial system, and substitution gives the displayed constants. Taking all degree- $\delta$ monomials and one relation yields Corollary 4.5 with the announced explicit bound.

Theorems 1.2 and 6.1Correct

The unlikely-intersection height bounds follow with the stated hypotheses

Pages 3 and 28–35 · Theorems 1.2 and 6.1 · arXiv:2607.09565v1

For a point on the line satisfying two independent multiplicative relations, the logarithmic G-functions produce local linear relations at every relevant place. At complex places, an integer combination cancels the $2\pi i$ terms; the two independent torus characters ensure that the resulting character is nonconstant. Multiplying the local factors gives a nonzero global relation of the asserted degree, and the explicit input estimates in Lemmas 6.9–6.12 satisfy Theorem 4.2 and Corollary 5.5.

Corollary 1.3Correct

The three-dimensional specialization is correct

Page 3 and page 35 · Corollary 1.3 and the end of Section 6.B.2 · arXiv:2607.09565v1

Distinct nonzero $a_1,a_2,a_3$ give the required line through the identity and the substitutions $n=3$ and $\lambda=4$ are correct. The two degree regimes use the appropriate clauses of Theorem 6.1, while the estimate for the number of complex places gives the stated dependence on $[K(s):\mathbb Q]$ .

Section 2.ATypo

The archimedean alternative in the definition of $\mathbb Q_v$ is mistyped

Page 5 · first paragraph of Section 2.A · arXiv:2607.09565v1

The sentence prints ' $\mathbb Q_v=\mathbb Q_{p_v}$ if $v$ is non-archimedean, or $\mathbb R$ if $v$ is non-archimedean.' The second occurrence must be 'archimedean.' This correction is uniquely fixed by $p_v=\infty$ at archimedean places, by the local-degree weights immediately below, and by every later use. No statement or proof is affected.

02Proofs4 reported findingsCorrect

The proofs of the explicit G-function bounds and the torus application are correct and complete. The only finding is a harmless notation typo in the preliminary place convention.

Theorem 4.2Correct and complete

Reduction to André's explicit inequality

Pages 18–22 · Proposition 4.6, Lemmas 4.7–4.8, and proof of Theorem 4.2 · arXiv:2607.09565v1

The paper corrects the relevant signs and normalizations in the cited inequality, checks the passage to monomials, and tracks the differential order, radii, height, and size terms. The choice of the auxiliary parameter satisfies the required lower bounds, and rearranging the final strict inequality gives exactly equation (4).

Proposition 6.8Correct and complete

Construction of a nonzero global relation

Pages 30–33 · Lemmas 6.5–6.7 and Proposition 6.8 · arXiv:2607.09565v1

At non-archimedean and real places, a character relation gives a homogeneous logarithmic relation directly. At each complex place the integer combination kills the integral multiple of $2\pi i$ without making the character constant. The product over complex places remains a nonzero polynomial and is a relation at every relevant place, with degree at most the stated archimedean-place count.

Section 6.BCorrect and complete

Explicit input data and final substitutions

Pages 33–35 · Lemmas 6.9–6.12 and Section 6.B.2 · arXiv:2607.09565v1

The singularities, generic radii, global radius, and size bounds are computed for the logarithmic system with the stated normalizations. These bounds meet the hypotheses of the global-relation estimates, and the calculations for the low- and high-degree regimes propagate all constants correctly to Theorem 6.1 and Corollary 1.3.

Section 2.ATypo

Local-field place label

Page 5 · definition of $\mathbb Q_v$ · arXiv:2607.09565v1

Replace the second 'non-archimedean' by 'archimedean.' The intended local field and every weighted-degree calculation are otherwise unambiguous.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.09565v1
Authors listed: Martin Orr
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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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. 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P(delta)

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P(delta)

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Explicit height bounds for G-functions and unlikely intersections with lines in tori

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The explicit G-function height bounds are correct

The unlikely-intersection height bounds follow with the stated hypotheses

The three-dimensional specialization is correct

The archimedean alternative in the definition of Qv\mathbb Q_vQv​ is mistyped

Reduction to André's explicit inequality

Construction of a nonzero global relation

Explicit input data and final substitutions

Local-field place label

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match

The archimedean alternative in the definition of $\mathbb Q_v$ is mistyped