Degree reduction under specialization

We examine the degree relationship between the elements of an ideal Isubset of or equal toR[x] and the elements of phi(I) where phi : R→R is a ring homomorphism. When R is a multivariate polynomial ring over a field, we use this relationship to show that the image of a Gröbner basis remains a Gröbner basis if we specialize all the variables but one, with no requirement on the dimension of I. As a corollary we obtain the GCD for a collection of parametric univariate polynomials. We also apply this result to solve parametric systems of polynomial equations and to reexamine the extension theorem for such systems.