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Smooth polynomials with several prescribed coefficients

László Mérai

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.

Smooth polynomials with several prescribed coefficients

Abstract

Let be the polynomial ring over the finite field of elements. A polynomial in is called -smooth (or -friable) if all its irreducible factors are of degree at most . In this paper, we investigate the distribution of -smooth (or -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.
Paper Structure (16 sections, 27 theorems, 189 equations)

This paper contains 16 sections, 27 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. Preliminaries
  4. Notations
  5. Arithmetically distributed relations
  6. Exponential sums
  7. Dickman's $\rho$ function
  8. Further preliminaries
  9. Distribution of smooth polynomials
  10. Analysis of ${\mathcal{S}}(\xi;n,m)$
  11. Major arc estimates
  12. Minor arc estimates
  13. Analysis of $S_{\mathcal{J}}$
  14. Proof of Theorem \ref{['thm:2']}
  15. Proof of Theorem \ref{['thm:1']}
  16. ...and 1 more sections

Key Result

Theorem 1

Let $0<\varepsilon<1/4$. Let ${\mathcal{I}}\subset\{0,1,\dots, n-1\}$ and $\alpha_i\in\mathbb{F}_q$ ($i\in {\mathcal{I}}$) such that $0\in {\mathcal{I}}$ and $\alpha_0\neq 0$. Assume, Write $\delta =\#{\mathcal{I}}/n$ and assume that Then for we have holds for some positive $C>0$ and $\eta>0$, where the implied constant may depend on $\varepsilon$.

Theorems & Definitions (37)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 27 more