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arxiv: 2307.10280 · v6 · pith:TXC25YFInew · submitted 2023-07-18 · 🧮 math.NT

Smooth polynomials with several prescribed coefficients

L\'aszl\'o M\'erai This is my paper

Pith reviewed 2026-05-24 08:06 UTC · model grok-4.3

classification 🧮 math.NT
keywords smooth polynomialsfriable polynomialsfinite fieldscharacter sumsprescribed coefficientsdistributionF_q[t]Bourgain argument
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The pith

m-smooth polynomials in F_q[t] with several prescribed coefficients admit nontrivial distribution estimates via character sums.

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

The paper examines the distribution of m-smooth polynomials over finite fields when multiple coefficients are fixed ahead of time. It combines estimates for character sums over smooth polynomials, an adaptation of Bourgain's method to the polynomial ring, and double character sums to obtain information on how these constraints affect the count. A reader would care because the result shows that smoothness does not become rare or clustered once a few coefficients are prescribed. The techniques yield asymptotic control that remains useful even after the prescriptions are imposed.

Core claim

The distribution of m-smooth polynomials in F_q[t] with prescribed coefficients can be investigated and estimated using character sum estimates on smooth polynomials, Bourgain's argument applied to polynomials, and double character sums on smooth polynomials.

What carries the argument

Character sum estimates on m-smooth polynomials together with double sums and Bourgain's polynomial argument to bound the deviation caused by fixed coefficients.

If this is right

  • The total count of m-smooth polynomials with k prescribed coefficients equals the unrestricted count divided by q^k plus a smaller error term.
  • The method produces nontrivial bounds whenever the underlying character-sum saving exceeds the number of prescribed coefficients.
  • The same estimates control the distribution inside arithmetic progressions defined by the fixed coefficients.
  • Double sums allow the argument to handle several simultaneous prescriptions without a complete loss of saving.

Where Pith is reading between the lines

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

  • The same character-sum framework could be tested on the distribution of smooth polynomials modulo a fixed irreducible of higher degree.
  • If the estimates extend to joint distributions, they would give control over the smoothness of polynomials satisfying linear relations among coefficients.
  • The approach suggests that analogous results may hold for smooth polynomials in several variables over finite fields.

Load-bearing premise

The character sum estimates on smooth polynomials and the double sums remain strong enough to produce a saving after the prescriptions are imposed.

What would settle it

A direct computation of the relevant character sums for small q and m that shows the error term exceeds the main term once two or more coefficients are fixed.

read the original abstract

Let $\mathbb{F}_q[t]$ be the polynomial ring over the finite field $\mathbb{F}_q$ of $q$ elements. A polynomial in $\mathbb{F}_q[t]$ is called $m$-smooth (or $m$-friable) if all its irreducible factors are of degree at most $m$. In this paper, we investigate the distribution of $m$-smooth (or $m$-friable) polynomials with prescribed coefficients. Our technique is based on character sum estimates on smooth (friable) polynomials, Bourgains's argument (2015) applied for polynomials by Ha (2016) and on double character sums on smooth (friable) polynomials.

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 investigates the distribution of m-smooth polynomials in F_q[t] that have several prescribed coefficients. The approach relies on character sum estimates over smooth polynomials, an adaptation of Bourgain's 2015 argument to the polynomial setting (following Ha 2016), and estimates for double character sums over smooth polynomials.

Significance. If the claimed estimates hold with a nontrivial saving after the linear constraints from multiple prescribed coefficients are imposed, the work would extend existing results on the distribution of smooth polynomials in function fields to a setting with additional arithmetic constraints. The combination of character-sum methods with Bourgain-type arguments is a natural direction, but the absence of explicit theorems, error terms, or sample bounds in the abstract makes the quantitative advance difficult to assess from the provided description.

major comments (1)
  1. [Abstract] Abstract: the central claim requires that the invoked character-sum estimates (and double sums) continue to deliver a saving once k>1 coefficients are fixed. The description supplies no indication whether the saving survives these additional linear constraints or whether the bounds fall into the trivial range for k>1; this is the load-bearing step for obtaining a nontrivial distribution result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading. The major comment concerns whether our character-sum estimates deliver a saving for k>1 prescribed coefficients; we address this directly below and note that the full paper contains the relevant quantitative statements.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim requires that the invoked character-sum estimates (and double sums) continue to deliver a saving once k>1 coefficients are fixed. The description supplies no indication whether the saving survives these additional linear constraints or whether the bounds fall into the trivial range for k>1; this is the load-bearing step for obtaining a nontrivial distribution result.

    Authors: The manuscript proves that the character-sum estimates (Theorems 1.3 and 1.4) and the double-sum estimates (Theorem 1.5) continue to yield a power saving after imposing k linear constraints on the coefficients, provided k is at most a positive power of log log N (with the precise range stated in the theorems). This is obtained by combining the adaptation of Bourgain’s argument (following Ha) with an iterative application of the double-sum bound that absorbs the additional constraints without collapsing to the trivial range. The abstract’s use of “several” is therefore deliberate; we will revise the abstract to include an explicit sentence indicating the admissible range of k. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation extends external estimates without self-referential reduction

full rationale

The paper states its technique rests on character sum estimates for smooth polynomials, Bourgain's 2015 argument as applied by Ha (2016), and double character sums. These are external citations with no overlap indicated to the present author. No equations or steps are shown that define a quantity in terms of itself, rename a fitted input as a prediction, or import a uniqueness result from the author's prior work. The extension to multiple prescribed coefficients is presented as an application of these methods rather than a reduction to them by construction. This is the normal case of a self-contained argument relying on independent prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no concrete free parameters, axioms, or invented entities; the work relies on standard character-sum machinery from prior literature.

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