Zero-sum-free tuples and hyperplane arrangements

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 $α_n^d$. We let $β_n^d$ 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 $α_n^d=ϕ(n)\binom{n-1}{d}$ when $d > n/2$, and in the general case, we prove recursive formulas, divisibility results, bounds, and asymptotic results for $α_n^d$ and $β_n^d$. In particular, $α_n^{n-1} = β_n^1= ϕ(n)$, suggesting that these sequences can be viewed as generalizations of Euler's totient function. We also relate the problem of computing $α_n^d$ 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 $α_n^d$ for all integers $n$ that are relatively prime to some determinants. We study the row and column patterns in the numbers $α_n^{d}$. We show that for any fixed $d$, $\{α_n^d \}$ is asymptotically equivalent to $\{ n^d\}$. We also show a connection between the asymptotic growth of $β_n^d$ and the value of the Riemann zeta function $ζ(d)$. Finally, we show that $α_n^d$ arises naturally in the study of Mathieu-Zhao subspaces in products of finite fields.