Vanishing ideals over rational parameterizations

Let R = K[y] = K[y1, . . . , yn] be a polynomial ring over an arbitrary field K and let F be a finite set {f1/g1, . . . , fs/gs} of rational functions in K(y), the quotient field of R, where fi (resp. gi) is in R (resp. R \ {0}) for all i. As usual we denote the affine and projective spaces over the field K by As and Ps−1, respectively. Points of the projective space Ps−1 are denoted by [α], where 0 6= α ∈ Ks. We consider the following sets parameterized by these rational functions: