Vanishing Ideals Over Odd Cycles

Let K be a finite field. Let X^* be a subset of the a ne space Kⁿ, which is parameterized by odd cycles. In this paper we give an explicit Gröbner 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.