Zero-sum-free tuples and hyperplane arrangements

TL;DR

This paper investigates zero-sum-free tuples in modular integer spaces, deriving formulas, bounds, and asymptotic behaviors, and connects these combinatorial structures to hyperplane arrangements and number theory concepts like Euler's totient and the Riemann zeta function.

Contribution

It provides recursive formulas, divisibility results, and asymptotic analysis for zero-sum-free tuples, linking combinatorics, hyperplane arrangements, and number theory.

Findings

lpha_n^d=(n)inom{n-1}{d} for d > n/2

lpha_n^{n-1}=eta_n^1=(n)

lpha_n^d is asymptotically equivalent to n^d

Abstract

A vector $(v_{1}, v_{2}, \cdots, v_{d})$ in $\mathbb{Z}_n^{d}$ is said to be a zero-sum-free $d$-tuple if there is no non-empty subset of its components whose sum is zero in $\mathbb{Z}_n$. We denote the cardinality of this collection by $\alpha_n^d$. We let $\beta_n^d$ denote the cardinality of the set of zero-sum-free tuples in $\mathbb{Z}_n^{d}$ where $\gcd(v_1, \cdots,v_d, n) = 1$. We show that $\alpha_n^d=\phi(n)\binom{n-1}{d}$ when $d > n/2$, and in the general case, we prove recursive formulas, divisibility results, bounds, and asymptotic results for $\alpha_n^d$ and $\beta_n^d$. In particular, $\alpha_n^{n-1} = \beta_n^1= \phi(n)$, suggesting that these sequences can be viewed as generalizations of Euler's totient function. We also relate the problem of computing $\alpha_n^d$ to counting points in the complement of a certain hyperplane arrangement defined over $\mathbb{Z}_n$. It is shown that the hyperplane arrangement's characteristic polynomial captures $\alpha_n^d$ for all integers $n$ that are relatively prime to some determinants. We study the row and column patterns in the numbers $\alpha_n^{d}$. We show that for any fixed $d$, $\{\alpha_n^d \}$ is asymptotically equivalent to $\{ n^d\}$. We also show a connection between the asymptotic growth of $\beta_n^d$ and the value of the Riemann zeta function $\zeta(d)$. Finally, we show that $\alpha_n^d$ arises naturally in the study of Mathieu-Zhao subspaces in products of finite fields.