  Motivated by the famous Skolem-Mahler-Lech theorem we initiate in this paper
the study of a natural class of determinantal varieties which we call {\em
Vandermonde varieties}. They are closely related to the varieties consisting of
all linear recurrence relations of a given order possessing a non-trivial
solution vanishing at a given set of integers. In the regular case, i.e., when
the dimension of a Vandermonde variety is the expected one, we present its free
resolution, obtain its degree and the Hilbert series. Some interesting
relations among Schur polynomials are derived. Many open problems and
conjectures are posed.
