Polynomial functions and spectral synthesis

Summary.

Polynomial functions on abelian groups are studied from the point of view of spectral synthesis. It is proved that the torsion free rank of an abelian group is finite if and only if each complex generalized polynomial on the group is a polynomial. A conjecture about spectral synthesis on abelian groups is formulated in terms of polynomial functions: spectral synthesis holds on an abelian group if and only if each complex bi-additive function on the group is a bilinear function of complex additive functions.