---
tag: C83KPT47R0
kind: concept
name: Gröbner basis
msc: [13P10, 68W30]
sources:
  - url: https://en.wikipedia.org/wiki/Gröbner_basis
same_as:
  wikidata: Q1551631
  wikipedia: Gröbner basis
---

## Summary

Fix a [[monomial order|E874NARWPV]] on a polynomial ring.
A Gröbner basis of an ideal is a finite generating set whose leading terms generate the ideal of all leading terms of the ideal.
With a Gröbner basis, the [[multivariate division algorithm|T8EN3CAEQF]] has a unique remainder.
So one can decide whether a polynomial lies in the ideal, and one can compute in the quotient ring.
Every ideal has a Gröbner basis, and [[Buchberger's algorithm|4RH1D40WG1]] computes one.
Gröbner bases are the basic tool of computational commutative algebra and algebraic geometry, including [[elimination theory|9PA8CVSWPD]].

## Explanations

### L1 human

For polynomials in one variable, an ideal is generated by one polynomial, and division by it decides membership.
In several variables, a set of generators may be badly chosen: a polynomial in the ideal can have a leading term that none of the generators' leading terms divides.
A Gröbner basis is a set of generators without this defect.
For example, $\set{x + y, x}$ generates the ideal $(x, y)$.
But $y = x - (x + y)$ is in the ideal, and for the order with $x > y$ its leading term $y$ is not divisible by $x$.
So this set is not a Gröbner basis, while the set $\set{x, y}$ is.