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A Generalization of Diophantine Tuples

Zijie Gu

TL;DR

This work generalizes Diophantine m-tuples to finite fields via d-f-Diophantine m-tuples with admissible polynomials f, and reduces counting such tuples to multiplicative character sums. Using the Weil bound and Shparlinski's method, it achieves power-saving error terms in the asymptotic count N_f^{binom([m]}{d)}(q) = q^m/(m! 2^{binom(m}{d)}) + O(q^{m−1/2}) for deg f ≥ 2 and O(q^{m−1}) for deg f = 1. The analysis highlights when Shparlinski's decoupling is effective and discusses limitations for more general symmetric polynomial conditions. Overall, the paper advances understanding of Diophantine-like structures in finite fields with sharper error terms than previous Lang-Weil-type bounds.

Abstract

This paper investigates a generalized version of Diophantine tuples in finite fields. Applying Shparlinski's method, we obtain power-saving results on the number of such tuples.

A Generalization of Diophantine Tuples

TL;DR

This work generalizes Diophantine m-tuples to finite fields via d-f-Diophantine m-tuples with admissible polynomials f, and reduces counting such tuples to multiplicative character sums. Using the Weil bound and Shparlinski's method, it achieves power-saving error terms in the asymptotic count N_f^{binom([m]}{d)}(q) = q^m/(m! 2^{binom(m}{d)}) + O(q^{m−1/2}) for deg f ≥ 2 and O(q^{m−1}) for deg f = 1. The analysis highlights when Shparlinski's decoupling is effective and discusses limitations for more general symmetric polynomial conditions. Overall, the paper advances understanding of Diophantine-like structures in finite fields with sharper error terms than previous Lang-Weil-type bounds.

Abstract

This paper investigates a generalized version of Diophantine tuples in finite fields. Applying Shparlinski's method, we obtain power-saving results on the number of such tuples.

Paper Structure

This paper contains 4 sections, 7 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. The Weil estimate
  3. Conversion from Diophantine Tuples to Character Sums
  4. Evaluations on $R(\varepsilon)$

Key Result

Theorem 1.5

The number $N_f^{\binom{[m]}{d}}(q)$ of $d$-$f$-Diophantine $m$-tuples, where $f$ is an admissible polynomial, satisfies Moreover, the implied constant depends on $\deg(f)$, $m$ and $d$ when $\deg(f)\geq 2$.

Theorems & Definitions (18)

  • Definition 1.3: Admissible polynomial
  • Definition 1.4
  • Theorem 1.5
  • Definition 2.1
  • Theorem 2.2: cf. zbMATH02121181
  • Claim 3.2
  • Lemma 4.1
  • proof
  • Remark 4.3
  • Lemma 4.4
  • ...and 8 more