Editing Hilbert's Nullstellensatz (section)

=== Using Gröbner bases ===

A [[Gröbner basis]] is an algorithmic concept that was introduced in 1973 by [[Bruno Buchberger]]. It is presently fundamental in [[computational geometry]]. A Gröbner basis is a special generating set of an ideal from which most properties of the ideal can easily be extracted. Those that are related to the Nullstellensatz are the following:
*An ideal contains {{math|1}} if and only if its [[reduced Gröbner basis]] (for any [[monomial ordering]]) is {{math|1}}.
*The number of the common zeros of the polynomials in a Gröbner basis is strongly related to the number of [[monomial]]s that are [[Gröbner basis#Reduction|irreducible]]s by the basis. Namely, the number of common zeros is infinite if and only if the same is true for the irreducible monomials; if the two numbers are finite, the number of irreducible monomials equals the numbers of zeros (in an algebraically closed field), counted with multiplicities.
*With a [[lexicographic order|lexicographic monomial order]], the common zeros can be computed by solving iteratively [[univariate polynomial]]s (this is not used in practice since one knows better algorithms).
* Strong Nullstellensatz: a power of {{mvar|p}} belongs to an ideal {{mvar|I}} if and only the [[Gröbner basis#Saturation|saturation]] of {{mvar|I}} by {{mvar|p}} produces the Gröbner basis {{math|1}}. Thus, the strong Nullstellensatz results almost immediately from the definition of the saturation.