Characterization of Classes of Polynomial Functions

In this paper some classes of local polynomial functions on abelian groups are characterized by the properties of their variety. For this characterization we introduce a numerical quantity depending on the variety of the local polynomial only. Moreover, we show that the known characterization of polynomials among generalized polynomials can be simplified: a generalized polynomial is a polynomial if and only if its variety contains finitely many linearly independent additive functions.