Degree reduction under specialization

We examine the degree relationship between the elements of an ideal I ⊆ R[x] and the elements of ’(I ) where ’ : R → R is a ring homomorphism. When R is a multivariate polynomial ring over a 3eld, we use this relationship to show that the image of a Gr4 obner basis remains a Gr4 obner 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 theoremfor such system s. c