Hilbert's Nullstellensatz

Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem"—see Satz) is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry, a branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert, who proved the Nullstellensatz and several other important related theorems named after him (like Hilbert's basis theorem).

With the notation common in algebraic geometry, the Nullstellensatz can also be formulated as

There are many known proofs of the theorem. One proof uses Zariski's lemma, which 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 this proof.[2]

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. 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 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.

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.[3] 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:[4]

If d is the maximum of the degrees of the fi, this bound may be simplified to

Kollár's result has been improved by several authors. As of 14 October 2012, the best improvement, due to M. Sombra is[5]

His bound improves Kollár's as soon as at least two of the degrees that are involved are lower than 3.