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arxiv: 2606.29688 · v1 · pith:5ZGQRTMWnew · submitted 2026-06-29 · 🧮 math.NT

Linear equations on t-modules

Yen-Tsung Chen , Wei-Cheng Huang , Changningphaabi Namoijam This is my paper

Pith reviewed 2026-06-30 05:41 UTC · model grok-4.3

classification 🧮 math.NT
keywords t-moduleslinear relationsFrobenius difference equationsglobal function fieldsF_q[t]-linear dependencealgorithmic computationalgebraic groups
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The pith

The F_q[t]-linear relations among points on a t-module correspond to polynomial solutions of Frobenius difference equations.

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

The paper investigates the determination of the group of F_q[t]-linear relations among finitely many points with L-coordinates on a d-dimensional t-module E defined over the global function field L. It establishes that these relations connect directly to the polynomial solutions of associated Frobenius difference equations. From this connection, an algorithm is derived to compute the entire module of linear relations. A reader would care because this provides a concrete computational method for an algebraic dependence problem that arises naturally in the arithmetic of function fields. The work parallels questions about linear relations in algebraic groups but operates in the t-module context.

Core claim

Let L be a global function field, and E be a d-dimensional t-module defined over L. Given finitely many points on E with entries in L, we establish the connection between their F_q[t]-linear relations and polynomial solutions of Frobenius difference equations. Consequently, we deduce an algorithm to compute the module of their F_q[t]-linear relations.

What carries the argument

The correspondence between F_q[t]-linear relations of L-points on a t-module and polynomial solutions of associated Frobenius difference equations.

If this is right

  • An algorithm computes the module of F_q[t]-linear relations for any such points and module.
  • This correspondence holds for arbitrary finite sets of L-rational points on any d-dimensional t-module.
  • The relations form a module that can be effectively determined via the difference equation solutions.

Where Pith is reading between the lines

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

  • The algorithm provides a decision procedure for whether any given vector of coefficients yields a relation.
  • The method may extend to computing relations among points on related structures such as Drinfeld modules.
  • Practical implementations could test the relations on explicit examples of low-dimensional t-modules.

Load-bearing premise

That the stated correspondence between linear relations and solutions of the Frobenius difference equations holds for arbitrary finitely many L-rational points on any d-dimensional t-module E defined over L, without further restrictions on the points or the module.

What would settle it

A counterexample consisting of a t-module E over L, points in E(L), and a linear relation over F_q[t] that is not captured by any polynomial solution to the corresponding Frobenius difference equation.

read the original abstract

Let $F$ be a number field. Given finitely many $F$-valued points on a commutative algebraic group defined over $F$, a question of interest to number theorists is the determination of the group of their linear relations. In this article, we investigate an analogous problem in the $t$-module setting. Let $L$ be a global function field, and $E$ be a $d$-dimensional $t$-module defined over $L$. Given finitely many points on $E$ with entries in $L$, we establish the connection between their $\mathbb{F}_q[t]$-linear relations and polynomial solutions of Frobenius difference equations. Consequently, we deduce an algorithm to compute the module of their $\mathbb{F}_q[t]$-linear relations.

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.

Referee Report

1 major / 0 minor

Summary. The paper claims that for a d-dimensional t-module E defined over a global function field L, the F_q[t]-linear relations among finitely many L-rational points on E correspond to polynomial solutions of associated Frobenius difference equations; from this correspondence it deduces an algorithm to compute the module of all such linear relations.

Significance. If the stated correspondence is valid without hidden restrictions on the points or the t-module, the work supplies an effective method for determining linear dependence in the t-module setting, paralleling classical results on linear relations for points on commutative algebraic groups over number fields. This would be a concrete computational advance in function-field arithmetic.

major comments (1)
  1. The central claim (connection between F_q[t]-linear relations and polynomial solutions of Frobenius difference equations, plus the resulting algorithm) is asserted for arbitrary finitely many L-rational points on any d-dimensional t-module E; the provided text supplies no derivation or list of hypotheses, so it is impossible to verify whether the correspondence holds in this generality or whether the algorithm terminates for all inputs.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for highlighting the need for greater clarity on hypotheses and derivations. We respond to the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The central claim (connection between F_q[t]-linear relations and polynomial solutions of Frobenius difference equations, plus the resulting algorithm) is asserted for arbitrary finitely many L-rational points on any d-dimensional t-module E; the provided text supplies no derivation or list of hypotheses, so it is impossible to verify whether the correspondence holds in this generality or whether the algorithm terminates for all inputs.

    Authors: The referee correctly observes that the submitted version states the main result in the abstract and introduction without a consolidated list of hypotheses or a self-contained derivation. The intended statement applies to any d-dimensional t-module E defined over the global function field L and any finite collection of L-rational points, with no further restrictions beyond the standard definition of t-modules. The correspondence is obtained by viewing the t-action as a Frobenius-linear endomorphism on the coordinate space and reducing F_q[t]-linear dependence among points to the kernel of an associated difference operator; the algorithm then enumerates polynomial solutions up to a degree bound determined by the dimension d. Because the current text does not make this explicit, we will add in the revision (i) an explicit hypotheses subsection at the start of Section 2 and (ii) a concise proof outline together with the termination argument in Section 3. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation appears self-contained

full rationale

The provided abstract states that the authors establish a connection between F_q[t]-linear relations on t-modules and polynomial solutions of Frobenius difference equations, then deduce an algorithm. No equations, fitted parameters, self-citations, or ansatzes are quoted that would reduce the claimed correspondence to a definition or prior input by construction. The reader's assessment similarly finds no indication of circularity in the abstract, and the skeptic notes the absence of full proof details that could reveal any reduction. This is the expected honest non-finding for a paper whose central claim is a mathematical correspondence without visible self-referential structure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no free parameters, invented entities, or non-standard axioms are visible. The result is stated to rest on the existing theory of t-modules and Frobenius actions over global function fields.

axioms (1)
  • domain assumption Standard properties of t-modules, their endomorphisms, and the Frobenius action over global function fields L.
    The connection and algorithm are asserted to follow from the established theory of t-modules.

pith-pipeline@v0.9.1-grok · 5660 in / 1088 out tokens · 33175 ms · 2026-06-30T05:41:30.155234+00:00 · methodology

discussion (0)

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Reference graph

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24 extracted references · 1 canonical work pages

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