Group Algebras of Finite Abelian Groups and their Applications to Combinatorial Problems

Abstract

Let G be an additive finite abelian group. In the last decades group algebras R[G] over suitable commutative rings R have turned out to be powerful tools for a growing variety of questions from combinatorics and number theory. Many of them can be reduced to the problem whether for some given sequence S = g1 · . . . · gl over G the elements f = (X1 − a1) · . . . · (Xl − al) 6= 0 ∈ R[G] for all a1, . . . , al ∈ R {0} . The present paper is devoted to this crucial problem. Before presenting our new results we recall the classical application of group algebras to the investigation of zero-sumfree sequences which is due to P. van Emde Boas, D. Kruyswijk and J.E. Olson (see [8], [9],[20]). Let d(G) denote the maximal length of a zero-sumfree sequence over G. Then d(G)+ 1 is the Davenport constant of G. For some commutative ring R, let d(G,R) denote the largest integer l ∈ N having the following property: there is some sequence S over G of length |S| = l such that (X1 − a1) · . . . · (Xl − al) 6= 0 ∈ R[G] for all a1, . . . , al ∈ R {0} . If S is zero-sumfree, R an integral domain, a1, . . . , al ∈ R {0} and