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An approach to the moments subset sum problem through systems of diagonal equations over finite fields

Juan Francisco Gottig, Mariana Pérez, Melina Privitelli

TL;DR

The paper addresses the m-th moment subset sum problem over finite fields by recasting it as a question about counting $\mathbb{F}_q$-rational points on systems of diagonal equations. It develops a unified algebraic-geometry framework, proving that the associated diagonal varieties are complete intersections and applying projective-closure techniques to derive explicit point-counting bounds with sharp dependence on $k,m$ and the exponents. These bounds yield concrete existence results for $N_m(k,\boldsymbol{b},D)$ in several regimes, including when $D=\mathbb{F}_q$ and when $D$ is the image of a polynomial, with notable refinements in the homogeneous and non-homogeneous cases and when $D=\{x^n:\,x\in\mathbb{F}_q\}$. The results improve prior work (e.g., Wan 2010, Marino 2020) by providing tighter error terms and broader parameter ranges, and they incorporate Brun sieve techniques to handle medium $k$ and large $m$, offering complementary existence criteria.

Abstract

Let $\mathbb{F}_q$ be the finite field of $q$ elements, for a given subset $D\subset \mathbb{F}_q$, $m\in \mathbb{N}$, an integer $k\leq |D|$ and $\boldsymbol{b}\in \mathbb{F}_q^m$ we are interested in determining the existence of a subset $S\subset D$ of cardinality $k$ such that $\sum_{a\in S}a^i=b_i$ for $i=1,\ldots, m$. This problem is known as the moment subset sum problem and it is $NP$-complete for a general $D$. We make a novel approach of this problem trough algebraic geometry tools analyzing the underlying variety and employing combinatorial techniques to estimate the number of $\mathbb{F}_q$-rational points on certain varieties. We managed to give estimates on the number of $\mathbb{F}_q$-rational points on certain diagonal equations and use this results to give estimations and existence results for the subset sum problem.

An approach to the moments subset sum problem through systems of diagonal equations over finite fields

TL;DR

The paper addresses the m-th moment subset sum problem over finite fields by recasting it as a question about counting -rational points on systems of diagonal equations. It develops a unified algebraic-geometry framework, proving that the associated diagonal varieties are complete intersections and applying projective-closure techniques to derive explicit point-counting bounds with sharp dependence on and the exponents. These bounds yield concrete existence results for in several regimes, including when and when is the image of a polynomial, with notable refinements in the homogeneous and non-homogeneous cases and when . The results improve prior work (e.g., Wan 2010, Marino 2020) by providing tighter error terms and broader parameter ranges, and they incorporate Brun sieve techniques to handle medium and large , offering complementary existence criteria.

Abstract

Let be the finite field of elements, for a given subset , , an integer and we are interested in determining the existence of a subset of cardinality such that for . This problem is known as the moment subset sum problem and it is -complete for a general . We make a novel approach of this problem trough algebraic geometry tools analyzing the underlying variety and employing combinatorial techniques to estimate the number of -rational points on certain varieties. We managed to give estimates on the number of -rational points on certain diagonal equations and use this results to give estimations and existence results for the subset sum problem.
Paper Structure (14 sections, 23 theorems, 103 equations)

This paper contains 14 sections, 23 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Basic notions of algebraic geometry
  3. Rational points
  4. Complete intersections
  5. Systems of diagonal equations
  6. The geometry of the projective closure
  7. Estimates on the number of $\mathbb F_{ q}$--rational solutions of systems of diagonal equations
  8. Homogeneous case
  9. Application to the subset sum problem
  10. $m$--th moment $k$-SSP for large $k$ and medium $m$
  11. $m$--th moment $k$-SSP for medium $k$ and large $m$
  12. Moment Subset Sums Problem with $D= \{f(x):\,\,x\in \mathbb F_{ q}\}$.
  13. Moment subset sums problem when $D=\{x^n: x\in \mathbb F_{ q}\}$
  14. Appendix A.

Key Result

Theorem 1.1

Wan2010 For all $\boldsymbol{b} \in \mathbb F_{ q}^m$,

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 37 more