Hilbert's Nullstellensatz

In mathematics, Hilbert's Nullstellensatz is a theorem that establishes a fundamental relationship between geometry and algebra. It was proven by David Hilbert in his second major paper on invariant theory in 1893 and became a foundational result of algebraic geometry.
There are several formulations of the Nullstellensatz, the most elementary of which deal with conditions for the existence of solutions to systems of multivariate polynomial equations over an algebraically closed field. The weak Nullstellensatz is a corollary of the Nullstellensatz which can be stated as follows. Consider a system of polynomial equations
in the variables, where are multivariate polynomials over an algebraically closed field. If the system does not have a solution, then there is "algebraic reason" for this situation: namely, this occurs precisely when there exist polynomials such that
Since the expression on the left-hand side must evaluate to 0 at any that solves the system of equations, it is obvious from the inconsistency that no solution can exist if this condition holds. Informally, the Nullstellensatz asserts that in the absence of such an inconsistency, a solution to the system of equations must exist.
The full Nullstellensatz is the following refinement: If is a polynomial such that every solution of the system of equations is also a solution of
then there is a similar type of "algebraic reason" for this occurrence: this occurs precisely when there exist a natural number and polynomials such that

Formulations

Let k be a field and K be an algebraically closed field extension of k. Consider the polynomial ring and let J be an ideal in this ring. The algebraic set defined by this ideal consists of all n-tuples in such that for all. Hilbert's Nullstellensatz states that if p is a polynomial in that vanishes on the algebraic set, i.e., for all, then there exists a natural number such that.
With the notation common in algebraic geometry, the Nullstellensatz can be formulated as
for every ideal J in with K algebraically closed. Here, denotes the radical of J, is the vanishing ideal of U, and is the zero locus of J. The assertion that is equivalent to the first formulation above with k = K algebraically closed, while the opposite inclusion is a straightforward consequence of the definitions.
An immediate corollary is the weak Nullstellensatz: If J is a proper ideal in, then is nonempty, i.e., for every algebraically closed extension, there exists a common zero in for all the polynomials in the ideal J. This is the reason for the name of the theorem, the full version of which can be proved easily from the 'weak' form using the Rabinowitsch trick. The assumption of considering common zeros in an algebraically closed field is essential here; for example, the elements of the proper ideal in do not have a common zero in
Specializing to the case of a single polynomial when and, one immediately recovers a restatement of the fundamental theorem of algebra: A polynomial P in has a root in if and only if. For this reason, the Nullstellensatz applied to can be thought of as a generalization of the fundamental theorem of algebra to systems of multivariable polynomial equations.
Taking K to be algebraically closed, the Nullstellensatz establishes an order-reversing bijective correspondence between the algebraic sets in and the radical ideals of In fact, more generally, one has a Galois connection between subsets of the space and subsets of the algebra, where "Zariski closure" and "radical of the ideal generated" are the closure operators.
As a particular example, consider an algebraic set consisting of a single point. Then is a maximal ideal. Conversely, every maximal ideal of the polynomial ring is of the form for some. This characterization of maximal ideals of polynomial rings over algebraically closed fields is another common formulation of the weak Nullstellensatz. As another example of this correspondence and a consequence of the Nullstellensatz, one can show that an algebraic subset W in is irreducible if and only if is a prime ideal.
More generally, for any ideal J in,
where the first intersection is taken over maximal ideals. This relationship is yet another common formulation of the Nullstellensatz.

Proofs

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 or as a linear combination of the generators of the ideal.

Using 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. If K is an algebraically closed field and is a maximal ideal of the ring of polynomials, then Zariski's lemma implies that is a finite field extension of K, and thus, by algebraic closure, must be K. From this, it follows that there is an such that for. In other words,
for some. But is clearly maximal, so. This is the weak Nullstellensatz: every maximal ideal of for algebraically closed K is of the form for some. Because of this close relationship, some texts refer to Zariski's lemma as the weak Nullstellensatz or as the 'algebraic version' of the weak Nullstellensatz.
The full Nullstellensatz can also be proved directly from Zariski's lemma without employing the Rabinowitsch trick. Here is a sketch of a proof using this lemma.
Let for algebraically closed field K, and let J be an ideal of A and be the common zeros of J in. Clearly,, where is the ideal of polynomials in A vanishing on V. To show the opposite inclusion, let. Then for some prime ideal in A. Let, where is the image of f under the natural map, and be a maximal ideal in R. By Zariski's lemma, is a finite extension of K, and thus, is K since K is algebraically closed. Let be the images of under the natural map. It follows that, by construction, but, so.

Using resultants

The following constructive proof of the weak form is one of the oldest proofs.
The resultant of two polynomials depending on a variable 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, every zero 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 that depends on, one chooses arbitrary values for the other variables. The fundamental theorem of algebra asserts that this choice can be extended to a zero of.
In the case of several polynomials a linear change of variables allows to suppose that is monic in the first variable. Then, one introduces new variables and one considers the resultant
As is in the ideal generated by the same is true for the coefficients in of the monomials in So, if 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.

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 if and only if its reduced Gröbner basis is.
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, counted with multiplicities.
With a lexicographic monomial order, the common zeros can be computed by solving iteratively univariate polynomials.
Strong Nullstellensatz: a power of belongs to an ideal if and only the saturation of by produces the Gröbner basis. Thus, the strong Nullstellensatz results almost immediately from the definition of the saturation.
Generalizations

The Nullstellensatz is subsumed by a systematic development of the theory of Jacobson rings, which are those rings in which every radical ideal is an intersection of maximal ideals. Given Zariski's lemma, proving the Nullstellensatz amounts to showing that if k is a field, then every finitely generated k-algebra R is Jacobson. More generally, one has the following theorem:
Other generalizations proceed from viewing the Nullstellensatz in scheme-theoretic terms as saying that for any field k and nonzero finitely generated k-algebra R, the morphism admits a section étale-locally. In this vein, one has the following theorem:
Serge Lang gave an extension of the Nullstellensatz to the case of infinitely many generators:

Effective Nullstellensatz

In all of its variants, Hilbert's Nullstellensatz asserts that some polynomial belongs or not to an ideal generated, say, by ; we have in the strong version, in the weak form. This means the existence or the non-existence of polynomials such that. The usual proofs of the Nullstellensatz are not constructive, non-effective, in the sense that they do not give any way to compute the.
It is thus a rather natural question to ask if there is an effective way to compute the or to prove that they do not exist. To solve this problem, it suffices to provide an upper bound on the total degree of the : such a bound reduces the problem to a finite system of linear equations that may be solved by usual linear algebra techniques. Any such upper bound is called an effective Nullstellensatz.
A related problem is the ideal membership problem, which consists in testing if a polynomial belongs to an ideal. For this problem also, a solution is provided by an upper bound on the degree of the. A general solution of the ideal membership problem provides an effective Nullstellensatz, at least for the weak form.
In 1925, Grete Hermann gave an upper bound for ideal membership problem that is doubly exponential in the number of variables. In 1982 Mayr and Meyer gave an example where the have a degree that is at least double exponential, showing that every general upper bound for the ideal membership problem is doubly exponential in the number of variables.
Since most mathematicians at the time assumed the effective Nullstellensatz was at least as hard as ideal membership, few mathematicians sought a bound better than double-exponential. In 1987, however, W. Dale Brownawell gave an upper bound for the effective Nullstellensatz that is simply exponential in the number of variables. Brownawell's proof relied on analytic techniques valid only in characteristic 0, but, one year later, János Kollár gave a purely algebraic proof, valid in any characteristic, of a slightly better bound.
In the case of the weak Nullstellensatz, Kollár's bound is the following:
If is the maximum of the degrees of the, this bound may be simplified to
An improvement due to M. Sombra is
His bound improves Kollár's as soon as at least two of the degrees that are involved are lower than 3.