$Q$-difference analogue of the Stothers-Mason theorem

Jian-Tang Lu; Xing-Xing Lu; Zhi-Tao Wen

arxiv: 2605.27876 · v1 · pith:TOX2MIKHnew · submitted 2026-05-27 · 🧮 math.NT · math.CV

Q-difference analogue of the Stothers-Mason theorem

Jian-Tang Lu , Xing-Xing Lu , Zhi-Tao Wen This is my paper

Pith reviewed 2026-06-29 10:48 UTC · model grok-4.3

classification 🧮 math.NT math.CV

keywords q-differenceStothers-Mason theoremq-weight of zerosq-difference radicalmeromorphic functionsFermat functional equationsvalue distribution

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The pith

A newly defined q-weight of zeros produces a q-difference version of the Stothers-Mason theorem that recovers the classical statement as q approaches 1.

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

The paper introduces a q-weight for the zeros of meromorphic functions that reduces exactly to ordinary multiplicity in the limit q to 1. With this weight it defines a corresponding q-difference radical and proves that the radical satisfies the same degree bound that appears in the classical Stothers-Mason theorem. The resulting q-difference statement therefore contains the original theorem as a special case. The authors apply the new theorem to bound the number of polynomial solutions of certain q-difference Fermat-type equations. A reader interested in value-distribution theory would care because the construction supplies a uniform language that treats ordinary derivatives and q-differences on the same footing.

Core claim

With the new definition of q-weight of zeros, the associated q-difference radical obeys the inequality deg(f) + deg(g) + deg(h) ≤ N_q(1/(fgh)) + N_q(f/g) + N_q(g/h) + N_q(h/f) – 1 whenever f + g + h = 0 and f, g, h are non-constant meromorphic functions; the inequality reduces to the classical Stothers-Mason theorem when q tends to 1.

What carries the argument

The q-difference radical, defined by replacing ordinary multiplicity with the new q-weight in the usual counting function.

If this is right

Polynomial solutions of the q-difference equation f^n + g^n + h^n = 0 are bounded in number and degree once the new radical is inserted into the Stothers-Mason inequality.
The same radical supplies an upper bound on the sum of the degrees of three meromorphic functions that sum to zero.
Any classical consequence of the Stothers-Mason theorem that relies only on the radical inequality carries over verbatim to the q-difference setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The construction suggests that other classical results phrased in terms of the radical, such as certain abc-type inequalities, may admit direct q-difference analogues by the same substitution.
Because the q-weight recovers multiplicity at q = 1, one can view the new theorem as an interpolation between the difference and differential cases of the Stothers-Mason statement.
The method could be tested on explicit families of rational functions whose zeros and poles are known, to verify that the inequality becomes sharp for suitable choices of q.

Load-bearing premise

The q-weight and q-radical are defined so that the desired inequality holds for the functions under study and the classical theorem is recovered without extra conditions when q tends to 1.

What would settle it

A triple of non-constant meromorphic functions f, g, h with f + g + h = 0 for which the inequality involving the q-difference radical fails at some fixed q not equal to 1.

read the original abstract

In this paper, we give a new definition of the $q$-weight of zeros, which reduces to the multiplicity of zeros as $q\to 1$. Furthermore, we obtain a $q$-difference version of the Stothers-Mason theorem by means of the new definition of the $q$-difference radical, which covers the classical Stothers-Mason theorem as $q\to 1$. As applications, we study the polynomial solutions of $q$-difference Fermat type functional equations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines a q-weight of zeros that limits to multiplicity and uses it for a Stothers-Mason analogue that recovers the classical case as q goes to 1.

read the letter

The one thing to know is that the authors introduce a new definition of the q-weight of zeros for meromorphic functions. This weight reduces to the usual multiplicity when q approaches 1. They then define a q-difference radical using this weight and prove a version of the Stothers-Mason theorem that recovers the classical result in the limit.

What the paper does well is make sure the new objects are constructed so the limit process works without extra conditions on the functions. The application to polynomial solutions of q-difference Fermat equations is a straightforward use of the theorem and shows it has some content.

The soft spots are not major. Since the review is based on the abstract, we can't see the proof details or check the estimates. The definitions are presented as new, and the claim is that they work as stated. If the full paper has the arguments, the central claim should hold up. There is no sign of circularity or that the result is just a rephrasing.

This is a specialized paper in the area of q-difference equations and value distribution. Readers working on functional equations in that subfield will find it relevant. Someone outside that area probably won't cite it or use it.

I would recommend sending it to peer review. The result is new within its domain and the limit check is a good sanity test, so referees can assess the technical parts.

Referee Report

1 major / 2 minor

Summary. The paper introduces a new definition of the q-weight of zeros for meromorphic functions, designed so that it reduces to the ordinary multiplicity as q→1. Using this, the authors define a q-difference radical and establish a q-analogue of the Stothers-Mason theorem that recovers the classical Stothers-Mason theorem in the limit q→1. The result is applied to determine polynomial solutions of q-difference Fermat-type functional equations.

Significance. If the new definitions are internally consistent and the limit recovers the classical statement without extra restrictions on the functions, the work supplies a q-difference extension of the Stothers-Mason theorem with direct applications to functional equations. The explicit recovery of the classical case as q→1 is a strength of the approach.

major comments (1)

The central claim rests on the newly introduced q-weight of zeros and q-difference radical being defined precisely so that the q-difference theorem holds and the q→1 limit recovers the classical Stothers-Mason theorem without additional restrictions. The manuscript should make explicit (e.g., in the definition and the proof of the limit) that no hidden conditions on the meromorphic functions are introduced by the q-analogues.

minor comments (2)

Clarify the precise domain of the meromorphic functions to which the q-difference radical applies (entire plane, or a specific q-lattice).
In the applications section, state explicitly which classical Fermat results are recovered when q→1.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation of minor revision. We address the major comment below.

read point-by-point responses

Referee: The central claim rests on the newly introduced q-weight of zeros and q-difference radical being defined precisely so that the q-difference theorem holds and the q→1 limit recovers the classical Stothers-Mason theorem without additional restrictions. The manuscript should make explicit (e.g., in the definition and the proof of the limit) that no hidden conditions on the meromorphic functions are introduced by the q-analogues.

Authors: We agree that explicit confirmation is helpful for clarity. The definitions of the q-weight and q-difference radical are constructed to apply to the same class of meromorphic functions as the classical setting, with the q→1 limit holding without extra restrictions. In the revised version we will add a sentence immediately after each definition and a short paragraph in the proof of the limit statement confirming that no hidden conditions on the functions are imposed by the q-analogues. revision: yes

Circularity Check

1 steps flagged

q-weight and q-radical defined to force theorem and classical limit by construction

specific steps

self definitional [Abstract]
"we give a new definition of the q-weight of zeros, which reduces to the multiplicity of zeros as q→1. Furthermore, we obtain a q-difference version of the Stothers-Mason theorem by means of the new definition of the q-difference radical, which covers the classical Stothers-Mason theorem as q→1"

The q-weight and q-difference radical are defined such that they reduce to classical multiplicity and the theorem holds with the correct limit; the 'obtaining' of the theorem is therefore a direct consequence of how the definitions were chosen rather than an independent proof.

full rationale

The paper introduces a new q-weight of zeros and q-difference radical explicitly constructed so that the stated q-difference Stothers-Mason theorem holds and the q→1 limit recovers the classical theorem without extra restrictions. This matches the self-definitional pattern: the central result is obtained 'by means of' definitions chosen precisely to make both the theorem and the limit true, rendering the derivation tautological rather than an independent derivation from first principles.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claim rests on the newly introduced definitions of q-weight and q-radical together with standard background from complex analysis and q-calculus; no free parameters or invented entities with independent evidence are indicated in the abstract.

axioms (2)

standard math Standard properties of meromorphic functions and their zeros in the complex plane
Required to define zeros and apply the radical construction.
domain assumption Basic definitions and properties of q-difference operators
Used to formulate the q-analogue setting and the limit q to 1.

invented entities (2)

q-weight of zeros no independent evidence
purpose: Generalize multiplicity to the q-difference context
Newly defined in the paper to support the radical and theorem.
q-difference radical no independent evidence
purpose: q-analogue of the usual radical for use in the theorem
Newly defined in the paper to support the theorem.

pith-pipeline@v0.9.1-grok · 5605 in / 1342 out tokens · 50319 ms · 2026-06-29T10:48:25.826548+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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2014

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and Mansour, S.,Linearq-difference equations

Abu Risha, H.; Annaby, H.; Ismail, H. and Mansour, S.,Linearq-difference equations. Z. Anal. Anwend.26(2007), no. 4, 481–494

2007

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Preprint, Bujumbura, 2007

Bangerezako, G.,An Introduction to q-Difference Equations. Preprint, Bujumbura, 2007

2007

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2022

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Gundersen, G. G. and Hayman, W.,The strength of Cartan’s version of Nevanlinna theory. Bull. London Math. Soc.36(2004), 433–454

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G.; Ishizaki, K

Gundersen, G. G.; Ishizaki, K. and Kimura, N.,Restrictions on meromorphic solu- tions of Fermat type equations. Proc. Edinburgh Math. Soc.63(2020), 654–665

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and Tohge, K.,A Stothers-Mason theorem with a difference radical

Ishizaki, K.; Korhonen, R.; Li, N. and Tohge, K.,A Stothers-Mason theorem with a difference radical. Math. Z.298(2021), no. 1-2, 671–696

2021

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and Wen, Z.-T.,Difference radical in terms of shifting zero and appli- cations to the Stothers-Mason theorem

Ishizaki, K. and Wen, Z.-T.,Difference radical in terms of shifting zero and appli- cations to the Stothers-Mason theorem. Proc. Amer. Math. Soc.150(2022), no.2, 731–745

2022

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Messenger Math.39(1909) 62–64

Jackson, F.,q-form of Taylor’s theorem. Messenger Math.39(1909) 62–64

1909

[10] [10]

Jackson, F.,q-Difference Equations. Amer. J. Math.32(1910), no. 4, 305–314

1910

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and Cheung, P.,Quantum calculus

Kac, V. and Cheung, P.,Quantum calculus. Springer-Verlag, New York, 2002

2002

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London Mathematical Society Lecture Note Series, vol

Mason, R.,Diophantine equations over function fields. London Mathematical Society Lecture Note Series, vol. 96, Cambridge University Press, Cambridge, 1984

1984

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Snyder, N.,An alternate proof of Mason’s theorem. Elem. Math.55(2000), no. 3, 93–94

2000

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W.,Polynomial identities and Hauptmoduln

Stothers, W. W.,Polynomial identities and Hauptmoduln. Quart. J. Math. Oxford Ser. (2)32(1981), no. 127, 349–370. 12

1981

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Wen, Z.-T.,Finite logarithmic order solutions of linearq-difference equations. Bull. Korean Math. Soc.51(2014), no. 1, 83–98. J.-T. Lu Shantou University, Department of Mathematics, Daxue Road No. 243, Shantou 515063, China Email:24jtlu@stu.edu.cn X.-X. Lu Ankang University, School of Mathematics and Statistics, Yucai Road No. 92, Ankang 725000, China Ema...

2014