We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 05 July 2011
Abstract
Gröbner bases have been an important tool for performing computations in polynomial rings since Buchberger first presented an algorithm to construct them in 1965. A generalized type of Gröbner basis was developed in the context of a valuation ring by Sweedler in 1988. The purpose of this paper is to concretize Sweedler's theory to the context of a k-subalgebra of a polynomial ring k[x1,…,xn], where k is a field and to present effective additional algorithms necessary for intrinsically constructing objects that we shall call SAGBI-Gröbner bases.
Introduction
SAGBI-Gröbner bases are simply the counterparts in ideals of subalgebras of polynomial rings to Gröbner bases of ideals of the full polynomial ring. Our intrinsic development of SAGBI-Gröbner theory will mimic that of ordinary Gröbner bases, an approach that has a certain aesthetic appeal. In addition, this parallel development provides another point of view from which to study the mechanics of Gröbner basis theory, which was first presented in an algorithmic fashion by Buchberger (1965) (see also Buchberger (1985)).
We assume little experience with Gröbner bases on the part of the reader; an overview of the subject may be found in the texts of Adams and Loustaunau (1994), Becker and Weispfennig (1993), and Cox, Little, and O'Shea (1992).
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
