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Expansion properties of polynomials over finite fields

Nuno Arala, Sam Chow

Abstract

We establish expansion properties for suitably generic polynomials of degree $d$ in $d+1$ variables over finite fields. In particular, we show that if $P\in\mathbb{F}_q[x_1,\ldots,x_{d+1}]$ is a polynomial of degree $d$ coming from an explicit, Zariski dense set, and $X_1,\ldots,X_{d+1}\subseteq\mathbb{F}_q$ are suitably large, then $|P(X_1,\ldots,X_{d+1})|=q-O(1)$. Our methods rely on a higher-degree extension of a result of Vinh on point--line incidences over a finite field.

Expansion properties of polynomials over finite fields

Abstract

We establish expansion properties for suitably generic polynomials of degree in variables over finite fields. In particular, we show that if is a polynomial of degree coming from an explicit, Zariski dense set, and are suitably large, then . Our methods rely on a higher-degree extension of a result of Vinh on point--line incidences over a finite field.
Paper Structure (4 sections, 10 theorems, 52 equations)

This paper contains 4 sections, 10 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Incidence geometry
  3. Proof of the main result
  4. Ternary polynomials of higher degree

Key Result

Theorem 1.2

Let $d\in\mathbb N$, and let $q$ be a prime power. Then there exists a constant $C>0$ such that, if $P\in\mathbb F_q[x_1,\ldots,x_{d+1}]$ is a nice polynomial of degree $d$, and $X_1,\ldots,X_{d+1}\subseteq\mathbb F_q$ satisfy $|X_k|\geq Cq^{d/(d+1)}$ for $k=1,2,\ldots,d+1$, then

Theorems & Definitions (18)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Theorem 1.7: Vinh
  • Theorem 2.1
  • proof
  • Lemma 3.1
  • ...and 8 more