# Gröbner Bases (Gröbner Bases)

## Description

In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring $K(x_1, \ldots ,x_n)$ over a field $K$. As an algorithmic problem, given a set of polynomials in $K(x_1, \ldots,x_n)$, determine a Gröbner basis.

## Parameters

$n$: number of variables in each polynomial

$d$: maximal total degree of the polynomials

## Table of Algorithms

Name Year Time Space Approximation Factor Model Reference
Buchberger's algorithm 1976 d^{({2}^{(n+o({1})})}) d^{({2}^{(n+o({1}))})}?? Exact Deterministic Time
Faugère F4 algorithm 1999 $O(C(n+D_{reg}, D_{reg})$^{\omega}) where omega is the exponent on matrix multiplication $O(C(n+D_{reg}, D_{reg})$^{2})? Exact Deterministic Time
Faugère F5 algorithm 2002 $O(C(n+D_{reg}, D_{reg})$^{\omega}) where omega is the exponent on matrix multiplication $O(C(n+D_{reg}, D_{reg})$^{2})? Exact Deterministic Time