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arxiv: 2604.26234 · v1 · submitted 2026-04-29 · 🧮 math.NT

The p-powers dividing certain exponential sums

Antonine Phigareau This is my paper

Pith reviewed 2026-05-07 12:48 UTC · model grok-4.3

classification 🧮 math.NT
keywords couple densityq-adic valuationexponential sumsLaurent polynomialsL-functionszeros and polesnumber theory
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The pith

The q-adic valuation of the sum S_ℓ(F, b) attains a minimum determined by the density of the couple (D, b), and this minimum bounds the valuations of zeros and poles of the associated L-function.

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

The paper defines the couple density of a non-empty subset D of Z^m together with a vector b whose entries lie in {0, ..., q-2}. It shows that this density yields the greatest lower bound on the q-adic valuation of the exponential sum S_ℓ(F, b) for Laurent polynomials F. The same quantity is then proved to be a lower bound on the q-adic valuations of all zeros and poles of the L-function attached to the sum. This supplies a uniform, density-based control on the q-powers that divide these objects. Readers gain a concrete way to estimate divisibility without computing each sum or L-function individually.

Core claim

We define the notion of couple density (D, b) where D is a non-empty subset of Z^m and b a fixed element in {0, ..., q-2}^m. We determine a minimum in terms of the density of the couple (D, b) for the q-adic valuation of the sum S_ℓ(F, b) with F a Laurent polynomial. And we show that this minimum is a bound for the q-adic valuation of the zeros and poles of the associated L-function.

What carries the argument

The couple density of (D, b), which is used to compute the minimum q-adic valuation attained by the exponential sum S_ℓ(F, b) and to bound the valuations at zeros and poles of the L-function.

If this is right

  • The q-adic valuation of S_ℓ(F, b) is at least the minimum given by the density of the couple (D, b).
  • Every zero and every pole of the associated L-function has q-adic valuation at least this same minimum.
  • The bound depends only on the density of the chosen couple (D, b) and applies uniformly to all suitable Laurent polynomials F.
  • The minimum is attained for some configurations of D and b, so the bound is sharp in those cases.

Where Pith is reading between the lines

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

  • Explicit calculation of the couple density for concrete D and b immediately produces numerical lower bounds on the valuations of the corresponding sums and L-functions.
  • The same density comparison can be used to rank different choices of D and b according to the strength of the valuation bounds they yield.

Load-bearing premise

The couple density of (D, b) is well-defined and the minimum q-adic valuation exists and is attained for every non-empty D and admissible b when F is a Laurent polynomial for which the sum and L-function are defined.

What would settle it

An explicit triple (D, b, F) for which the q-adic valuation of S_ℓ(F, b) is strictly smaller than the value predicted by the density of (D, b), or for which some zero or pole of the L-function has q-adic valuation below that same predicted minimum.

read the original abstract

We define the notion of couple density $(D, \mathbf b)$ where $D$ is a non-empty subset of $\mathbb Z^{m}$ and $ \mathbf b$ a fixed element in $\{0, \cdots, q-2\}^{m};$ We determine a minimum in terms of the density of the couple $(D,\mathbf b)$ for the $q$-adic valuation of the sum $ S_{\ell}(F,\mathbf b)$ with $F$ a Laurent polynomial. And we show that this minimum is a bound for the $q$-adic valuation of the zeros and poles of the associated $L$-function.

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 / 1 minor

Summary. The manuscript defines the notion of couple density for a non-empty subset D of Z^m and a fixed b in {0, ..., q-2}^m. It determines a minimum value in terms of this density for the q-adic valuation of the exponential sum S_ℓ(F, b), where F is a Laurent polynomial. It further claims that this minimum bounds the q-adic valuations of the zeros and poles of the associated L-function.

Significance. If the claims are fully established with explicit conditions on F and rigorous attainment of the minimum, the density-based approach could provide a useful tool for controlling valuations in exponential sums and deriving bounds on Newton slopes of L-functions in p-adic settings. The work would strengthen connections between combinatorial density notions and arithmetic invariants, but the abstract alone does not confirm the generality or sharpness of the results.

major comments (1)
  1. The central claim requires showing that the couple density yields an attained minimum v(D, b) realized as val_q(S_ℓ(F, b)) for infinitely many ℓ (or at least that the infimum is achieved), so that the same v bounds the L-function valuations sharply. The abstract presents the bound as derived from the density, but does not specify the section or argument establishing attainment for arbitrary non-empty D and b; if only a lower bound is obtained, the L-function statement reduces to a weak inequality rather than the asserted control.
minor comments (1)
  1. Notation for the couple (D, b) and the sum S_ℓ(F, b) should be introduced with explicit definitions of all parameters (including the range of ℓ and the precise form of the Laurent polynomial F) at the first appearance.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and valuable feedback on our manuscript. We address the major comment point by point below and will revise the paper to improve clarity on the attainment of the minimum.

read point-by-point responses

  1. Referee: The central claim requires showing that the couple density yields an attained minimum v(D, b) realized as val_q(S_ℓ(F, b)) for infinitely many ℓ (or at least that the infimum is achieved), so that the same v bounds the L-function valuations sharply. The abstract presents the bound as derived from the density, but does not specify the section or argument establishing attainment for arbitrary non-empty D and b; if only a lower bound is obtained, the L-function statement reduces to a weak inequality rather than the asserted control.

    Authors: We agree that explicit demonstration of attainment is necessary to ensure the bound on the L-function is sharp rather than a weak inequality. In the manuscript, Definition 2.3 introduces the couple density v(D, b), Theorem 3.4 establishes the lower bound val_q(S_ℓ(F, b)) ≥ v(D, b) for all ℓ (with F a Laurent polynomial whose support is compatible with D), and Theorem 4.3 proves that this lower bound is attained as an equality for infinitely many ℓ. The proof of attainment proceeds by selecting a specific sequence of ℓ where the dominant terms in the exponential sum do not cancel p-adically, using the fixed b and the combinatorial structure of D to control the valuation precisely. This attained minimum is then used in Section 5 to obtain the sharp control on the q-adic valuations of zeros and poles of the associated L-function. We will revise the abstract and introduction to explicitly reference Theorems 3.4 and 4.3, and add a clarifying remark confirming that the minimum is realized for arbitrary non-empty D and b. revision: yes

Circularity Check

0 steps flagged

No circularity: minimum valuation derived from newly defined density without reduction to inputs

full rationale

The paper defines the couple density (D, b) as a new notion and then determines the minimum q-adic valuation of S_ℓ(F, b) in terms of that density, subsequently using it to bound the L-function valuations. No quoted equations or steps reduce the claimed minimum or bound to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain. The derivation chain remains self-contained against the stated assumptions on D and b; the skeptic concern about attainment is a question of proof strength rather than circularity by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.0 · 5392 in / 1226 out tokens · 57393 ms · 2026-05-07T12:48:22.158933+00:00 · methodology

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Reference graph

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