On a Theorem of J. E. Olson and an Application (Vanishing Sums in Finite Abelianp-Groups)

Generalizing a theorem by J. E. Olson determining the Davenportʹs constant of a finite abelianp-groupA, we prove that ifS1, . . . ,Skare given sets of integers satisfying suitable conditions and ifg1, . . . ,gk A, then a nontrivial vanishing sum of the forms1g1+ • • • +skgk,si Simay be found. The result may be applied to the problem of finding polynomials inFp[x] having small degree and prescribed value set onFp.