  Let K be a finite field. Let X* be a subset of the affine space Kn, which is
parameterized by odd cycles. In this paper we give an explicit Gr\"obner basis
for the vanishing ideal, I(X*), of X*. We give an explicit formula for the
regularity of I(X*) and finally if X* is parameterized by an odd cycle of
length k, we show that the Hilbert function of the vanishing ideal of X* can be
written as linear combination of Hilbert functions of degenerate torus.
