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

Martin Orr

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

T0 review · 0 major / 6 minor · reviewed 2026-07-13 · grok-4.5

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Pith's one-line read Explicit height bounds for unexpected G-function relations give concrete height control on unlikely intersections of lines through the identity with rank-1 subgroups of tori.

desk verdict Solid explicit constants for Bombieri–André plus the first clean written G-function application to tori; restrictions are real but openly stated and do not break the claims. read the letter →

arxiv 2607.09565 v1 pith:I5PGR6XK submitted 2026-07-10 math.NT

Martin Orr This is my paper · ORCID

classification math.NT MSC 11G3011G5011J91

keywords G-functionsheightboundsunlikelyintersectionstoriBombieri–Andréprincipleglobalrelationsexplicitconstants

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper turns Bombieri–André’s asymptotic height bound for points with non-trivial global polynomial relations among G-function values into fully explicit inequalities, with constants written in terms of the number of functions, their sizes, radii of convergence, and the singularities of their differential equation. It then specialises the machinery to the logarithms of linear forms 1+a_i X, which are G-functions, and thereby obtains an explicit polynomial height bound for points of a line C through the identity in G_m^n that lie in the union of algebraic subgroups of codimension at least two. The bound depends on the number of complex places of the field of definition of the point and on the height of the coefficients of the line; for small Galois degree it improves the previously known explicit bound of Habegger, while for large degree it is weaker because the degree of the constructed global relation grows with the number of complex embeddings. The work also supplies the first fully written application of the G-functions method to unlikely intersections inside algebraic groups rather than Shimura varieties.

What carries the argument

The corrected explicit inequality of André (Proposition 4.6) that converts a collection of linearly independent local relations of degree 1 into a global height lower bound; combined with the construction of an (r_v)-global relation of degree at most max{τ(K(s)),1} obtained by multiplying the linear forms attached to independent multiplicative characters at the complex places (Proposition 6.8).

What would settle it

Compute the actual Weil height of a concrete point of a line 1+a_i ξ in G_m^3 that lies in a rank-1 subgroup and has Galois degree 4 or 5; if that height exceeds the numerical constant 2708(H+1) of Corollary 1.3, the claimed bound is false.

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Extended reading notes

Core claim

All points at which a super-strongly non-trivial (or merely non-trivial) (r_v)-global polynomial relation of degree δ holds among a fixed tuple of G-functions satisfy an explicit height bound of the shape 8 s λ^3/κ^2 (log(δ)+O(1)) (σ_y+ρ) (or the refined super-strong form involving Hilbert-function ratios and binomial constants c_6(η), c_7(η)). Specialised to the n+1 G-functions {1, log(1+a_i X)} attached to a line through the identity in G_m^n, this yields an explicit height bound for every point of the line that lies in Σ_2, polynomial of degree n in the number of complex places of its field of definition.

Load-bearing premise

The curve must pass through the identity (or a torsion point) so that the local relations at complex places can be taken with rational coefficients and still work at all non-archimedean and real places; without that the degree of the global relation cannot be controlled solely by the number of complex embeddings.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Martin has finally written down the explicit constants that everyone has been waving their hands about for Bombieri–André, and then used them on the simplest possible unlikely-intersection problem in tori. That is the news.

What is new: Corollaries 4.5, 5.4 and 5.5 give fully explicit c1 (and improved c2 under super-strong non-triviality) for the height of points carrying non-trivial global relations among G-functions. Theorem 4.2 is the refined engine that tracks multiple independent relations and the ambient monomial space. The torus application (Theorem 6.1 / Corollary 1.3) is the first place the method has been written out for algebraic groups rather than Shimura varieties; for lines through the identity he gets a polynomial-in-degree bound that beats Habegger’s enormous constant for Galois degrees up to roughly 10^4 (or far higher if you take Habegger’s general-curve constant). The comparison in §6.C is honest.

He does the bookkeeping carefully: corrects the two small errors in André’s VII.3.5 inequality (s = max{1,#Sin} and the good-apparent-singularity hypothesis), introduces (rv)-global relations so the radii can be smaller than the radii of convergence without an extra ad-hoc step, and computes every input (radii, sizes, singularities of the log system) explicitly for the series log(1+aiX). Ax–Schanuel gives the functional independence, Hilbert-function bounds are standard, and the construction of the global relation of degree ≤ max{\tau(K(s)),1} is clean once you accept the geometric restrictions.

Soft spots are exactly the ones he flags: the bound depends on [K(s):Q], only Σ2 is treated, the curve must pass through a torsion point, and the ambient variety must be a line. Those are essential to getting Q-coefficients and controlled degree; they are not hidden. The paper does not claim more. No free parameters, no circularity.

This is for people who actually want to run the G-function machine or who need concrete numbers for small-degree points on lines. It deserves a serious referee; the math is checkable line-by-line. I would cite the explicit corollaries and the torus illustration.

Referee Report

0 major / 6 minor

Summary. The paper gives fully explicit constants in Bombieri–André height bounds for points at which G-functions satisfy non-trivial (or super-strongly non-trivial) global polynomial relations. After correcting minor errors in André’s inequality (Proposition 4.6), it proves refined bounds (Theorems 4.2, 5.1–5.2 and Corollaries 4.4–4.5, 5.4–5.5) that track the number of independent relations, the Hilbert function of the ideal of functional relations, and an acceptable system of radii (r_v). These are applied to lines C through the identity in G_m^n that are not contained in a proper algebraic subgroup: for s in C(K) ∩ Σ_2 one obtains an explicit height bound for the local parameter x(s) that is polynomial in δ = max{1, τ(K(s))} and linear in a height H of the coefficients of the line (Theorem 6.1 and Corollary 1.3). The input data (singularities, radii, sizes of the logarithmic G-functions and of the associated differential operator) are computed in Lemmas 6.9–6.12, and the (r_v)-global relations of controlled degree are constructed in Proposition 6.8 via multiplicative characters.

Significance. Explicit constants for the Bombieri–André principle have been missing for decades; the paper supplies them in several usable forms and demonstrates that the G-function method yields competitive height bounds for low-degree points on lines in tori (better than Habegger’s bound for [K(s):Q] up to roughly 10^4 even after optimising Habegger for lines). The introduction of (r_v)-global relations cleanly handles the usual radius mismatch and produces sharper constants. The application is the first written instance of the method for algebraic groups rather than Shimura varieties, and the inexplicit existence argument in §6.A is a useful pedagogical illustration. All constants are derived from first principles with no free parameters; the geometric restrictions (line through a torsion point, Σ_2 only) are essential to the method and are stated openly.

minor comments (6)

Abstract and title: “polymomial” should be “polynomial”.
§2.A: the sentence defining Q_v for archimedean places reads “or R if v is non-archimedean”; the second occurrence should be “archimedean”.
§3.A, Remark 3.1: the discussion of the erroneous definition in André is helpful but could be shortened; the corrected characterisation via Lemma 3.2 is the one actually used later.
§4.C: the two corrections to André’s inequality are carefully justified; a short parenthetical note that the original asymptotic statements remain valid would reassure readers who only need the non-explicit theory.
§6.C: the numerical comparison with Habegger is clear, but the optimised constant 2^{27 n^{17}} for lines is stated without a reference or short derivation; a one-sentence sketch or citation would help.
Throughout: the notation σ(Ξ) for the height of a finite set is standard, yet a brief reminder that it coincides with the usual affine logarithmic height of the corresponding vector would aid non-specialists.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: explicit bounds follow from a corrected André inequality plus self-contained constructions of radii, sizes and (r_v)-global relations; self-citations are background only.

full rationale

The load-bearing chain is: (i) correct André’s inequality (Prop. 4.6, fixing the #Sin=0 and bad-apparent-singularity cases with explicit counter-examples), (ii) deduce the refined height bound Thm. 4.2 by linear algebra on monomials and Nakayama, (iii) obtain the super-strong versions via Hilbert-function multiplication (Thms. 5.1–5.2, Cor. 5.5), (iv) for the tori application construct the G-functions log(1+a_i X), prove homogeneous algebraic independence by Ax–Schanuel (Lemma 6.2), build (r_v)-global relations of degree au(K(s)) from multiplicative characters (Prop. 6.8), and compute the concrete input data ho, ho_y, ho_ heta, s, H from the residue matrices and p-adic radii of log (Lemmas 6.9–6.12). All steps are first-principles calculations or external classical theorems; no parameter is fitted to data and then re-used as a prediction, and the self-citations (DO21, DO25) appear only as motivational background for the G-functions method, never as the justification of a uniqueness claim or of the numerical constants. The geometric restrictions (line through the identity, Σ_2 only) are openly stated as essential to the method and do not create an internal definitional loop. Hence the circularity score is at most 1.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The paper works entirely inside classical Diophantine geometry and the theory of G-functions. The only external black boxes are André’s intermediate inequality (corrected in the paper), the Ax–Schanuel theorem for functional independence of logarithms, and standard facts about Hilbert functions of homogeneous ideals. No free parameters are fitted; all constants are derived. The mild generalisation “(r_v)-global relations” is introduced for technical convenience but is not an ontological novelty.

assumptions (4)

domain assumption André’s inequality [And89, VII, 3.5 Prop.] (after the two corrections stated in Proposition 4.6) relating the sum of local heights of a point to the sizes of the G-functions and the differential operator.
Used as the sole quantitative engine for all height bounds in §§4–5; the paper corrects two errors but does not re-prove the inequality from scratch.
standard math Ax–Schanuel theorem [Ax71, Thm. 1] implying that the logarithms F_i = log(1+a_i X) are homogeneously algebraically independent over Q(X) when the line is not contained in a proper subgroup.
Invoked in Lemma 6.2 to guarantee that the constructed global relations are non-trivial (hence super-strongly non-trivial).
standard math Chardin’s upper bound and Nesterenko’s lower bound for Hilbert functions of homogeneous prime ideals (Lemma 5.9).
Used to convert dimension/degree data into concrete binomial-coefficient estimates that feed Theorem 5.2.
standard math Standard properties of absolute Weil heights, Gauss norms and radii of convergence of power series over number fields (normalisations of §2).
Background language for all size and radius estimates.

invented entities (1)

(r_v)-global relations independent evidence
purpose: Mild generalisation of ordinary global relations that allows the radii of the constructed relations to be smaller than the radii of convergence of the G-functions, simplifying the bookkeeping in applications.
Introduced in §3.D; every ordinary global relation is (r_v)-global, so the notion does not enlarge the set of points under consideration, only the flexibility of the proofs.

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Pith. "Pith review of Explicit height bounds for G-functions and unlikely intersections with lines in tori." pith.science (2026). https://pith.science/paper/I5PGR6XK

@misc{pith2026260709565,
  author       = {Pith},
  title        = {Pith review of: Explicit height bounds for G-functions and unlikely intersections with lines in tori},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I5PGR6XK}},
  note         = {Machine review of arXiv:2607.09565}
}

read the original abstract

This paper computes explicit constants in Bombieri and Andr\'e'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.

Discussion (0). Continue with ORCID to comment.

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[1] [1]

Andr\' e , G -functions and geometry , Aspects of Mathematics, E13, Friedr

Y. Andr\' e , G -functions and geometry , Aspects of Mathematics, E13, Friedr. Vieweg & Sohn, Braunschweig, 1989

1989

[2] [2]

Andr\'e, G -functions, motives, and unlikely intersections---old and new , Atti Accad

Y. Andr\'e, G -functions, motives, and unlikely intersections---old and new , Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 36 (2025), no. 3, 381--392

2025

[3] [3]

Ax, On S chanuel's conjectures , Ann

J. Ax, On S chanuel's conjectures , Ann. of Math. (2) 93 (1971), 252--268

1971

[4] [4]

M. A. Barkatou and S. S. Maddah, Removing apparent singularities of systems of linear differential equations with rational function coefficients, I SSAC '15--- P roceedings of the 2015 ACM I nternational S ymposium on S ymbolic and A lgebraic C omputation, ACM, New York, 2015, pp. 53--60

2015

[5] [5]

Binyamini and D

G. Binyamini and D. Masser, Effective Andr\'e Oort for non-compact curves in Hilbert modular varieties , Comptes Rendus. Math\'ematique 359 (2021), no. 3, 313--321

2021

[6] [6]

Bombieri, D

E. Bombieri, D. Masser, and U. Zannier, Intersecting a curve with algebraic subgroups of multiplicative groups, Internat. Math. Res. Notices (1999), no. 20, 1119--1140

1999

[7] [7]

Bombieri, On G -functions , Recent progress in analytic number theory, V ol

E. Bombieri, On G -functions , Recent progress in analytic number theory, V ol. 2 ( D urham, 1979), Academic Press, London-New York, 1981, pp. 1--67

1979

[8] [8]

Chardin, Une majoration de la fonction de H ilbert et ses cons\'equences pour l'interpolation alg\'ebrique , Bull

M. Chardin, Une majoration de la fonction de H ilbert et ses cons\'equences pour l'interpolation alg\'ebrique , Bull. Soc. Math. France 117 (1989), no. 3, 305--318

1989