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 $\alpha_n^d$. 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.