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).
, one immediately recovers a restatement of the fundamental theorem of algebra: a polynomial P in
For this reason, the (weak) Nullstellensatz has been referred to as a generalization of the fundamental theorem of algebra for multivariable polynomials.
then V(I) cannot be empty, i.e. there exists a common zero for all the polynomials in the ideal in every algebraically closed extension of k. 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 (X2 + 1) in
With the notation common in algebraic geometry, the Nullstellensatz can also be formulated as for every ideal J.
denotes the radical of J and I(U) is the ideal of all polynomials that vanish on the set U.
we obtain an order-reversing bijective correspondence between the algebraic sets in Kn 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.
More generally, Conversely, every maximal ideal of the polynomial ring
As another example, an algebraic subset W in Kn is irreducible (in the Zariski topology) if and only if
Others are constructive, as based on algorithms for expressing 1 or pr as a linear combination of the generators of the ideal.
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
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.
Those that are related to the Nullstellensatz are the following: 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 (necessarily of the form
In this vein, one has the following theorem: Serge Lang gave an extension of the Nullstellensatz to the case of infinitely many generators: In all of its variants, Hilbert's Nullstellensatz asserts that some polynomial g belongs or not to an ideal generated, say, by f1, ..., fk; we have g = f r in the strong version, g = 1 in the weak form.
This means the existence or the non-existence of polynomials g1, ..., gk such that g = f1g1 + ... + fkgk.
The usual proofs of the Nullstellensatz are not constructive, non-effective, in the sense that they do not give any way to compute the gi.
It is thus a rather natural question to ask if there is an effective way to compute the gi (and the exponent r in the strong form) 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 gi: such a bound reduces the problem to a finite system of linear equations that may be solved by usual linear algebra techniques.
For this problem also, a solution is provided by an upper bound on the degree of the gi.
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 gi 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.
In 1987, however, W. Dale Brownawell gave an upper bound for the effective Nullstellensatz that is simply exponential in the number of variables.
[8] 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.
at, say, the origin can be shown to be a Noetherian local ring that is a unique factorization domain.