We examine the degree relationship between the elements of an ideal I ⊆R[x] and the elements of f(I ) where f: 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 Greobner basis remains a Greobner 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 arametric univariate polynomials. We also apply this result to solve parametric systems of polynomial equations and to reexamine the extension theorem for such systems.
Degree Reduction under Specialization
FORTUNA, ELISABETTA;GIANNI, PATRIZIA;
2001-01-01
Abstract
We examine the degree relationship between the elements of an ideal I ⊆R[x] and the elements of f(I ) where f: 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 Greobner basis remains a Greobner 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 arametric univariate polynomials. We also apply this result to solve parametric systems of polynomial equations and to reexamine the extension theorem for such systems.File in questo prodotto:
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