THE INTERPOLATION PROBLEM FOR K-SPARSE POLYNOMIALS AND CHARACTER SUMS

Motivated by a remarkable result of D. Y. Grigoriev and M. Karpinski, the interpolation problem for k-sparse multivariate polynomials has received some attention in recent years. In this note we want to show that essentially all of the results obtained so far hold more generally for k-sparse sums of characters of abelian monoids, thereby providing a useful unified approach to this active field of research. As it turns out, there are basically two different situations, in the first one reduction to the (rather trivial) case of cyclic monoids is possible, in the general case we can handle direct products of abelian monoids by using informations about the factors. Basic ingredients of these approaches are the construction of distinction sets for characters and zero-test sets for k-sparse character sums. © 1991.