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

Time Complexity Graph

Space Complexity Graph

Time-Space Tradeoff