Improved bounds for zero-sum cycles in Zₚᵈ

For a finite Abelian group (, +), let n () denote the smallest positive integer n such that for each labelling of the arcs of the complete digraph of order n using elements from, there exists a directed cycle such that the total sum of the arc-labels along the cycle equals 0. Alon and Krivelevich initiated the study of the parameter n () on cyclic groups and proved that n (Zq) =O (q q). Studying the prototypical case when =Zₚᵈ is a power of a cyclic group of prime order, Letzter and Morrison recently showed that n (Zₚᵈ) O (pd (d) ²) and that n (Z₂ᵈ) O (d d). They then posed the problem of proving an (asymptotically optimal) upper bound of n (Zₚᵈ) O (pd) for all primes p and d N. In this paper, we solve this problem for p=2 and improve their bound for all primes p 3 by proving n (Z₂ᵈ) 5d and n (Zₚᵈ) O (pd d). While the first bound determines n (Z₂ᵈ) up to a multiplicative error of 5, the second bound is tight up to a d factor. Moreover, our result shows that a tight bound of n (Zₚᵈ) = (pd) for arbitrary p and d would follow from a (strong form) of the well-known conjecture of Jaeger, Linial, Payan and Tarsi on additive bases in Zₚᵈ. Along the way to proving these results, we establish a generalization of a hypergraph matching result by Haxell in a matroidal setting. Concretely, we obtain sufficient conditions for the existence of matchings in a hypergraph whose hyperedges are labelled by the elements of a matroid, with the property that the edges in the matching induce a basis of the matroid. We believe that these statements are of independent interest.