Zero-sum-free tuples and hyperplane arrangements
Sunil K. Chebolu, Papa A. Sissokho

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 in is said to be a zero-sum-free -tuple if there is no non-empty subset of its components whose sum is zero in . We denote the cardinality of this collection by . We let denote the cardinality of the set of zero-sum-free tuples in where . We show that when , and in the general case, we prove recursive formulas, divisibility results, bounds, and asymptotic results for and . In particular, , suggesting that these sequences can be viewed as generalizations of Euler's totient function. We also relate the problem of computing to counting points in the complement of a certain hyperplane arrangement defined over . It…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Point processes and geometric inequalities · Advanced Mathematical Identities
