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$f$-Diophantine sets over finite fields via quasi-random hypergraphs from multivariate polynomials

Seoyoung Kim, Chi Hoi Yip, Semin Yoo

TL;DR

This work studies $f$-Diophantine sets over finite fields by constructing explicit quasi-random $k$-uniform hypergraphs $Y_{f,q}$ from multivariate polynomials. The authors define admissible polynomials and prove a uniform edge-count bound $N^*_{Y_{f,q}}({\mathcal O}_k^e)=\frac{q^{2k}}{2}+O(q^{2k-1})$, establishing quasi-randomness and enabling precise counting results. They derive an asymptotic formula for the number of $f$-Diophantine sets of size $m$ and obtain effective Lang-Weil–type refinements, with immediate corollaries for Paley hypergraphs and for $k$-Diophantine $m$-tuples over ${\mathbb F}_q$, including a complete asymptotic count for the Diophantine tuples when $m\ge k$. Overall, the paper provides a unified, polynomial-based approach to quasi-random hypergraphs that yields new counting results with sharpened error terms, advancing the intersection of combinatorics and arithmetic geometry in finite fields.

Abstract

We investigate $f$-Diophantine sets over finite fields via new explicit constructions of families of quasi-random hypergraphs from multivariate polynomials. In particular, our construction not only offers a systematic method for constructing quasi-random hypergraphs but also provides a unified framework for studying various hypergraphs arising from multivariate polynomials over finite fields, including Paley sum hypergraphs, and hypergraphs derived from Diophantine tuples and their generalizations. We derive an asymptotic formula for the number of $k$-Diophantine $m$-tuples, answering a question of Hammonds et al., and study some related questions for $f$-Diophantine sets, extending and improving several recent works. We also sharpen a classical estimate of Chung and Graham on even partial octahedrons in Paley sum hypergraphs.

$f$-Diophantine sets over finite fields via quasi-random hypergraphs from multivariate polynomials

TL;DR

This work studies -Diophantine sets over finite fields by constructing explicit quasi-random -uniform hypergraphs from multivariate polynomials. The authors define admissible polynomials and prove a uniform edge-count bound , establishing quasi-randomness and enabling precise counting results. They derive an asymptotic formula for the number of -Diophantine sets of size and obtain effective Lang-Weil–type refinements, with immediate corollaries for Paley hypergraphs and for -Diophantine -tuples over , including a complete asymptotic count for the Diophantine tuples when . Overall, the paper provides a unified, polynomial-based approach to quasi-random hypergraphs that yields new counting results with sharpened error terms, advancing the intersection of combinatorics and arithmetic geometry in finite fields.

Abstract

We investigate -Diophantine sets over finite fields via new explicit constructions of families of quasi-random hypergraphs from multivariate polynomials. In particular, our construction not only offers a systematic method for constructing quasi-random hypergraphs but also provides a unified framework for studying various hypergraphs arising from multivariate polynomials over finite fields, including Paley sum hypergraphs, and hypergraphs derived from Diophantine tuples and their generalizations. We derive an asymptotic formula for the number of -Diophantine -tuples, answering a question of Hammonds et al., and study some related questions for -Diophantine sets, extending and improving several recent works. We also sharpen a classical estimate of Chung and Graham on even partial octahedrons in Paley sum hypergraphs.

Paper Structure

This paper contains 10 sections, 12 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Quasi-random hypergraphs
  3. Main results
  4. Applications to Diophantine tuples over finite fields
  5. More backgrounds and motivations
  6. A key proposition
  7. Proof of \ref{['thm:main']}
  8. A sufficient condition
  9. Proof of \ref{['thm:main']} via \ref{['prop:key']}
  10. Asymptotic formula for the number of $f$-Diophantine sets with size $m$

Key Result

Theorem 1.3

Let $k\geq 2$ and let $d\geq 1$. Let $\mathcal{Y}_{d}$ be the family consisting of all hypergraphs $Y_{f,q}$, where $q$ is an odd prime power and $f \in {\mathbb F}_q[x_1,x_2,\ldots,x_k]$ is an admissible polynomial with degree $d$. Then holds uniformly for all $Y_{f,q} \in \mathcal{Y}_{d}$. In particular, $\mathcal{Y}_{d}$ is a family of quasi-random $k$-uniform hypergraphs.

Theorems & Definitions (26)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Lemma 2.1: Schwartz–Zippel lemma
  • Remark 2.2
  • ...and 16 more