Editing Algebraic geometry (section)

===Gröbner basis===

{{main|Gröbner basis}}

A [[Gröbner basis]] is a system of [[Ideal (ring theory)#Definitions and motivation|generators]] of a polynomial [[ideal (ring theory)|ideal]] whose computation allows the deduction of many properties of the affine algebraic variety defined by the ideal.

Given an ideal ''I'' defining an algebraic set ''V'':
* ''V'' is empty (over an algebraically closed extension of the basis field), if and only if the Gröbner basis for any [[monomial order]]ing is reduced to {1}.
* By means of the [[Hilbert series]] one may compute the [[dimension of an algebraic variety|dimension]] and the [[degree of an algebraic variety|degree]] of ''V'' from any Gröbner basis of ''I'' for a monomial ordering refining the total degree.
* If the dimension of ''V'' is 0, one may compute the points (finite in number) of ''V'' from any Gröbner basis of ''I'' (see [[Systems of polynomial equations]]).
* A Gröbner basis computation allows one to remove from ''V'' all irreducible components which are contained in a given hypersurface.
* A Gröbner basis computation allows one to compute the Zariski closure of the image of ''V'' by the projection on the ''k'' first coordinates, and the subset of the image where the projection is not [[proper morphism|proper]].
* More generally Gröbner basis computations allow one to compute the Zariski closure of the image and the [[critical point (mathematics)|critical point]]s of a rational function of ''V'' into another affine variety.

Gröbner basis computations do not allow one to compute directly the [[primary decomposition]] of ''I'' nor the prime ideals defining the irreducible components of ''V'', but most algorithms for this involve Gröbner basis computation. The algorithms which are not based on Gröbner bases use [[regular chain]]s but may need Gröbner bases in some exceptional situations.

Gröbner bases are deemed to be difficult to compute. In fact they may contain, in the worst case, polynomials whose degree is doubly exponential in the number of variables and a number of polynomials which is also doubly exponential. However, this is only a worst case complexity, and the complexity bound of Lazard's algorithm of 1979 may frequently apply. [[Faugère F5 algorithm]] realizes this complexity, as it may be viewed as an improvement of Lazard's 1979 algorithm. It follows that the best implementations allow one to compute almost routinely with algebraic sets of degree more than 100. This means that, presently, the difficulty of computing a Gröbner basis is strongly related to the intrinsic difficulty of the problem.