Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Estimates for short character sums evaluated at homogeneous polynomials

Rena Chu

TL;DR

This work develops nontrivial Burgess-type bounds for short Dirichlet character sums evaluated at a broad class of homogeneous polynomials in arbitrary dimensions. The authors reduce polynomial arguments to norm forms via a linear change of variables, lift characters to finite-field extensions, and exploit a multiplicative energy framework together with lattice geometry to control redundancies and high-moment sums. A central innovation is a recursive, energy-based approach that simultaneously reduces box size and dimension, yielding bounds that reach the Burgess threshold p^{1/4+κ} for many forms and dimensions, and extending previous results beyond forms that split over F_p. The combination of norm-form lifting, lattice-based energy estimates, and Burgess-type amplification yields concrete, explicit savings and broad applicability to short-sum problems tied to Diophantine counting and beyond. This framework sharpens our understanding of how polynomial structure interacts with multiplicative characters in short-sum regimes and provides tools potentially adaptable to broader arithmetic-geometry contexts.

Abstract

Let $p$ be a prime. We prove bounds on short Dirichlet character sums evaluated at a class of homogeneous polynomials in arbitrary dimensions. In every dimension, this bound is nontrivial for sums over boxes with side lengths as short as $p^{1/4 + κ}$ for any $κ>0$. Our methods capitalize on the relationship between characters mod $p$ and characters over finite field extensions as well as bounds on the multiplicative energy of sets in products of finite fields.

Estimates for short character sums evaluated at homogeneous polynomials

TL;DR

This work develops nontrivial Burgess-type bounds for short Dirichlet character sums evaluated at a broad class of homogeneous polynomials in arbitrary dimensions. The authors reduce polynomial arguments to norm forms via a linear change of variables, lift characters to finite-field extensions, and exploit a multiplicative energy framework together with lattice geometry to control redundancies and high-moment sums. A central innovation is a recursive, energy-based approach that simultaneously reduces box size and dimension, yielding bounds that reach the Burgess threshold p^{1/4+κ} for many forms and dimensions, and extending previous results beyond forms that split over F_p. The combination of norm-form lifting, lattice-based energy estimates, and Burgess-type amplification yields concrete, explicit savings and broad applicability to short-sum problems tied to Diophantine counting and beyond. This framework sharpens our understanding of how polynomial structure interacts with multiplicative characters in short-sum regimes and provides tools potentially adaptable to broader arithmetic-geometry contexts.

Abstract

Let be a prime. We prove bounds on short Dirichlet character sums evaluated at a class of homogeneous polynomials in arbitrary dimensions. In every dimension, this bound is nontrivial for sums over boxes with side lengths as short as for any . Our methods capitalize on the relationship between characters mod and characters over finite field extensions as well as bounds on the multiplicative energy of sets in products of finite fields.
Paper Structure (37 sections, 33 theorems, 257 equations)

This paper contains 37 sections, 33 theorems, 257 equations.

Table of Contents

  1. Introduction
  2. Previous literature
  3. Results on character sums over finite fields
  4. Results on character sums evaluated at forms
  5. An overview of the method
  6. The strength of Theorem \ref{['thm:main']}
  7. The cases $n\leq k$ and $k\leq n$
  8. Outline of the paper
  9. Notation
  10. Norm forms and lifting of characters
  11. Lifting characters mod $p$
  12. A key change of variables
  13. Deduction of Theorem \ref{['thm:main2']} from Theorem \ref{['cor:main_energy']}
  14. The initial setup
  15. Reducing to boxes with comparable side lengths
  16. ...and 22 more sections

Key Result

Theorem 1.1

Fix a prime $p$ and an integer $n\geq 1$, and let $F\in \mathbb{F}_p[X_1,...,X_n]$ be a form in $n$ variables and of degree $n$. Suppose $F$ splits into linear forms over the algebraic closure $\overline{\mathbb{F}}_p$, where the linear forms are linearly independent over $\overline{\mathbb{F}}_p$,

Theorems & Definitions (53)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Lemma 2.1: Fla53
  • Proposition 2.2
  • ...and 43 more