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Relation between algebraic varieties and polynomial ideals
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros", or more literally, "zero locus theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert, who proved the Nullstellensatz in his second major paper on invariant theory in 1893 (following his seminal 1890 paper in which he proved Hilbert's basis theorem).

There are many known proofs of the theorem. Some are non constructive, such as the first one. Others are constructive, as based on algorithms for expressing 1 or as a linear combination of the generators of the ideal.

Zariski's lemma asserts that if a field is finitely generated as an associative algebra over a field k, then it is a finite field extension of k (that is, it is also finitely generated as a vector space). Here is a sketch of a proof using this lemma. Let (k algebraically closed field), I an ideal of A, and V the common zeros of I in Clearly, Let Then for some prime ideal in A. Let and a maximal ideal in By Zariski's lemma, is a finite extension of k; thus, is k since k is algebraically closed. Let be the images of under the natural map passing through It follows that and .

The following constructive proof of the weak form is one of the oldest proofs (the strong form results from the Rabinowitsch trick, which is also constructive). The resultant of two polynomials depending on a variable x and other variables is a polynomial in the other variables that is in the ideal generated by the two polynomials, and has the following properties: if one of the polynomials is monic in x, every zero (in the other variables) of the resultant may be extended into a common zero of the two polynomials. The proof is as follows. If the ideal is principal, generated by a non constant polynomial p that depends on x, one chooses arbitrary values for the other variables. The fundamental theorem of algebra asserts that this choice can be extended to a zero of p.

In the case of several polynomials a linear change of variables allows to suppose that is monic in the first variable x. Then, one introduces new variables and one considers the resultant

As R is in the ideal generated by the same is true for the coefficients in R of the monomials in So, if 1 is in the ideal generated by these coefficients, it is also in the ideal generated by On the other hand, if these coefficients have a common zero, this zero can be extended to a common zero of by the above property of the resultant. This proves the weak Nullstellensatz by induction on the number of variables.

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 1 if and only if its reduced Gröbner basis (for any monomial ordering) is 1. The number of the common zeros of the polynomials in a Gröbner basis is strongly related to the number of monomials that are irreducibles 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 monomial order, the common zeros can be computed by solving iteratively univariate polynomials (this is not used in practice since one knows better algorithms). Strong Nullstellensatz: a power of p belongs to an ideal I if and only the saturation of I by p produces the Gröbner basis 1. Thus, the strong Nullstellensatz results almost immediately from the definition of the saturation.