Zero-sum-free tuples and hyperplane arrangements

A vector (v1, v2,⋯, vd) in Zn is said to be a zero-sum-free d-tuple if there is no nonempty subset of its components whose sum is zero in Zn. We denote the cardinality of this collection by αn. We let β d n denote the cardinality of the set of zero-sumfree tuples in Zn where gcd(v1,⋯, vd, n) = 1. We show that αn = φ(n)( n−1 d ) when d > n/2, and in the general case, we prove recursive formulas, divisibility results, bounds, and asymptotic results for αn and β d n. In particular, α n−1 n = β n = φ(n), suggesting that these sequences can be viewed as generalizations of Euler’s totient function. We also relate the problem of computing αn to counting points in the complement of a certain hyperplane arrangement defined over Zn. It is shown that the hyperplane arrangement’s characteristic polynomial captures αn for all integers n that are relatively prime to some determinants. We study the row and column patterns in the numbers αn. We show that for any fixed d, {αn} is asymptotically equivalent to {n}. We also show a connection between the asymptotic growth of β n and the value of the Riemann zeta function ζ(d). Finally, we show that αn arises naturally in the study of Mathieu-Zhao subspaces in products of finite fields.