Equidistribution of polynomial sequences in function fields: resolution of a conjecture

J\'er\'emy Champagne; Th\'ai Ho\`ang L\^e; Trevor D. Wooley; Yu-Ru Liu; Zhenchao Ge

arxiv: 2512.16118 · v3 · pith:IRVJD2MTnew · submitted 2025-12-18 · 🧮 math.NT

Equidistribution of polynomial sequences in function fields: resolution of a conjecture

J\'er\'emy Champagne , Zhenchao Ge , Th\'ai Ho\`ang L\^e , Yu-Ru Liu , Trevor D. Wooley This is my paper

Pith reviewed 2026-05-22 13:02 UTC · model grok-4.3

classification 🧮 math.NT

keywords equidistributionpolynomial sequencesfunction fieldsLaurent seriesfinite fieldsirrationalitynumber theory

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The pith

A polynomial with coefficients in Laurent series is equidistributed in the torus over finite fields when it has an irrational coefficient at a degree k not divisible by the characteristic p and K excludes all p-power multiples of k.

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

The paper establishes equidistribution for the sequence of values of the polynomial sum α_r u^r in the quotient T = K_∞ / F_q[t] as u runs through all polynomials over the finite field F_q of characteristic p. The result requires that at least one coefficient α_k is irrational in the Laurent series field and that the support set K of degrees contains neither k itself if p divides k nor any higher multiple p^v k. This fully resolves a conjecture previously posed by three of the authors. A sympathetic reader cares because the theorem supplies the precise function-field counterpart to classical results on equidistribution of polynomial sequences, identifying exactly when the values become dense and uniform in the non-archimedean torus.

Core claim

The polynomial ∑_{r∈K∪{0}} α_r u^r is equidistributed in T whenever α_k is irrational for some k∈K satisfying p∤k, and also p^v k ∉ K for any positive integer v. This conclusion resolves in full a conjecture made jointly by the third, fourth and fifth authors.

What carries the argument

The finite support set K of positive integers together with the exclusion of all p-power multiples of any admissible degree k, combined with the definition of irrationality of coefficients inside the Laurent series field K_∞.

If this is right

The sequence of polynomial values fills the torus T uniformly with respect to the natural Haar measure.
Equidistribution holds for every finite linear combination of powers whose degrees satisfy the stated non-multiplicity condition by p.
The result completes the classification of equidistributed polynomial sequences in this function-field setting.
The same criterion applies verbatim to any finite field F_q of characteristic p.

Where Pith is reading between the lines

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

The same exclusion condition on K is likely necessary, because its violation produces algebraic relations that confine the sequence to a lower-dimensional subtorus.
Techniques used here may adapt directly to equidistribution questions for polynomials over local fields of positive characteristic or to higher-dimensional tori.
Boundary cases where p divides k but no higher multiple appears in K could be tested numerically to determine whether equidistribution still holds in a weaker sense.
The result suggests a parallel classification problem for sequences in p-adic integers where similar characteristic-dependent dependencies arise.

Load-bearing premise

The support set K must contain no p-power multiple of the degree k at which the coefficient is irrational; without this exclusion the values may lie in a proper subgroup due to linear dependence over the field of characteristic p.

What would settle it

Explicit computation of the closure of the sequence for a concrete K containing p k (with p∤k and α_k irrational) that shows the image is contained in a coset of a proper subgroup of T rather than being dense.

read the original abstract

Let $\mathbb F_q$ be the finite field of $q$ elements having characteristic $p$, and denote by $\mathbb K_\infty=\mathbb F_q((1/t))$ the field of formal Laurent series in $1/t$. We consider the equidistribution in $\mathbb T=\mathbb K_\infty/\mathbb F_q[t]$ of the values of polynomials $f(u)\in \mathbb K_\infty [u]$ as $u$ varies over $\mathbb F_q[t]$. Let $\mathcal K$ be a finite set of positive integers, and suppose that $\alpha_r\in \mathbb K_\infty$ for $r\in \mathcal K\cup \{0\}$. We show that the polynomial $\sum_{r\in \mathcal K\cup\{0\}}\alpha_ru^r$ is equidistributed in $\mathbb T$ whenever $\alpha_k$ is irrational for some $k\in \mathcal K$ satisfying $p\nmid k$, and also $p^vk\not\in \mathcal K$ for any positive integer $v$. This conclusion resolves in full a conjecture made jointly by the third, fourth and fifth authors.

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

This paper fully resolves the conjecture on equidistribution of polynomial sequences in function fields under precise conditions.

read the letter

This paper resolves the conjecture on equidistribution of polynomial sequences in function fields. The authors deliver the complete statement under the conditions they state in the abstract. What is new is the full resolution: the polynomial is equidistributed whenever there is some k in K with alpha_k irrational, p does not divide k, and no higher power p^v k is in K for v positive. This prevents the sequence from being trapped in a proper closed subgroup of T due to the characteristic p. The paper does well in setting up the problem with the field K_infty and the torus T, then applying harmonic analysis to prove the result. The condition on the set K is carefully chosen to match the cases where equidistribution holds. The soft spots are minor. The definition of irrationality for elements in K_infty is standard, but the proof must carefully handle the cases where the exclusion holds. From what is visible, the argument does not have internal contradictions or unsupported steps. This work is for researchers in analytic number theory who work with function fields as analogs for Diophantine problems. Readers who follow results on equidistribution in positive characteristic or who might use this for further analogies will find it useful. The paper shows honest engagement with the conjecture and the literature. It deserves a serious referee. I recommend that it go through peer review.

Referee Report

0 major / 2 minor

Summary. The paper proves an equidistribution result for polynomial sequences over function fields. Let F_q be a finite field of characteristic p and K_infty = F_q((1/t)). For a finite set K of positive integers and coefficients alpha_r in K_infty, the values of the polynomial sum_{r in K union {0}} alpha_r u^r are shown to be equidistributed in the compact group T = K_infty / F_q[t] as u varies over F_q[t], provided there exists k in K with p not dividing k such that alpha_k is irrational and p^v k not in K for any positive integer v. This resolves a conjecture of the third, fourth, and fifth authors.

Significance. If the result holds, it provides a complete resolution of the stated conjecture in the function-field setting. The argument relies on standard harmonic analysis on the compact abelian group T and identifies a precise exclusion condition on K that blocks reductions to proper closed subgroups via the freshman's dream in characteristic p. The derivation is parameter-free and directly addresses the conjecture without auxiliary assumptions or fitted constants.

minor comments (2)

§1: The precise formulation of the original conjecture (as stated by the third, fourth and fifth authors) could be quoted verbatim to make the resolution claim fully self-contained.
Definition of irrationality for elements of K_infty: a brief reminder of the standard definition (non-membership in F_q(t)) would aid readers unfamiliar with the function-field setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, accurate summary of the main result, and recommendation to accept the manuscript. The report correctly identifies the key exclusion condition on the set K that prevents reductions via the freshman's dream and confirms that the argument resolves the conjecture without additional assumptions.

Circularity Check

0 steps flagged

No significant circularity; direct proof of equidistribution via harmonic analysis

full rationale

The paper establishes equidistribution of the polynomial sum over the finite set K union {0} in the compact group T by applying standard harmonic analysis techniques under explicit hypotheses on the coefficients alpha_r and the exclusion of p^v k from K. This resolves the stated conjecture but does so through a self-contained argument that does not reduce any claimed prediction to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The irrationality condition and characteristic-p exclusion are used directly to ensure the image is dense rather than confined to a proper subgroup, with no renaming of known results or smuggling of ansatzes via prior work. The derivation stands independently against external benchmarks in function field equidistribution.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The argument relies on standard properties of the completion K_∞, the Haar measure on T, and Weyl's criterion for equidistribution; no new entities or fitted parameters are introduced.

axioms (2)

standard math K_∞ is the completion of F_q(t) with respect to the valuation at infinity, and T carries its normalized Haar measure.
Invoked to define the ambient space and the notion of equidistribution.
domain assumption Irrationality of α_k means α_k lies outside the subfield F_q(t).
Central to the hypothesis; standard in function-field Diophantine approximation.

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discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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work page arXiv

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Bergelson and A

V. Bergelson and A. Leibman,A Weyl-type equidistribution theorem in finite characteristic, Adv. Math.289(2016), 928–950

work page 2016

[3] [3]

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work page 1952

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Dijksma,Uniform distribution of polynomials overGF{q, x}inGF[q, x]

A. Dijksma,Uniform distribution of polynomials overGF{q, x}inGF[q, x]. II, Indag. Math.32 (1970), 187–195

work page 1970

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R. M. Kubota,Waring’s problem forF q[x], Dissertationes Math. (Rozprawy Mat.)117(1974), 60 pp

work page 1974

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T. H. Lˆ e, Y.-R. Liu and T. D. Wooley,Equidistribution of polynomial sequences in function fields, with applications, Adv. Math.479(2025), article no. 110424, 44pp

work page 2025

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Weil,Basic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 144, Springer-Verlag New York, New York, 1967

A. Weil,Basic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 144, Springer-Verlag New York, New York, 1967

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work page 1916