Hostname: page-component-797576ffbb-cx6qr Total loading time: 0 Render date: 2023-12-11T20:27:39.994Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "useRatesEcommerce": true } hasContentIssue false

# 4 - Gröbner Bases and the Buchberger Algorithm

Published online by Cambridge University Press:  06 July 2010

## Summary

This chapter gives a “look under the hood” at the algorithm that actually lets us perform computations over a polynomial ring. In order to work with polynomials, we need to be able to answer the ideal membership question. For example, there is no chance of writing down a minimal free resolution if we cannot even find a minimal set of generators for an ideal. How might we do this? If R = k[x], then the Euclidean algorithm allows us to solve the problem. What makes things work is that there is an invariant (degree), and a process which reduces the invariant. Then ideal membership can be decided by the division algorithm. When we run the univariate division algorithm, we “divide into” the initial (or lead) term. In the multivariate case we'll have to come up with some notion of initial term – for example, what is the initial term of x2y + y2x? It turns out that this means we have to produce an ordering of the monomials of R = k[x1, …, xn]. This is pretty straightforward. Unfortunately, we will find that even once we have a division algorithm in place, we still cannot solve the question of ideal membership. The missing piece is a multivariate analog of the Euclidean algorithm, which gave us a good set of generators (one!) in the univariate case. But there is a simple and beautiful solution to our difficulty; the Buchberger algorithm is a systematic way of producing a set of generators (a Gröbner basis) for an ideal or module over R so that the division algorithm works.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2003

## Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

# Save book to 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.

Available formats
×

# Save book 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 Dropbox.

Available formats
×

# Save book to Google Drive

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.

Available formats
×