“Hilbert's Nullstellensatz”的意思、由来-开放百科全书

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).

Formulation

Let k be a field (such as the rational numbers) and K be an algebraically closed field extension (such as the complex numbers), consider the polynomial ring

and let I be an ideal in this ring. The algebraic set V(I) defined by this ideal consists of all n-tuples x = (x₁,...,x_n) in Kⁿ such that f(x) = 0 for all f in I. Hilbert's Nullstellensatz states that if p is some polynomial in

that vanishes on the algebraic set V(I), i.e. p(x) = 0 for all x in V(I), then there exists a natural number r such that p^r is in I.

An immediate corollary is the "weak Nullstellensatz": The ideal

contains 1 if and only if the polynomials in I do not have any common zeros in Kⁿ. It may also be formulated as follows: if I is a proper ideal in

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, 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

do not have a common zero in

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

for every ideal J. Here,

denotes the radical of J and I(U) is the ideal of all polynomials that vanish on the set U.

In this way, 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.

Conversely, every maximal ideal of the polynomial ring

(note that

is algebraically closed) is of the form

for some

As another example, an algebraic subset W in Kⁿ is irreducible (in the Zariski topology) if and only if

is a prime ideal.

Proof and generalization

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.^[1]

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

. It follows that

and

The Nullstellensatz will also follow trivially once one systematically developed the theory of a Jacobson ring, a ring in which a radical ideal is an intersection of maximal ideals. Let

be a Jacobson ring. If

is a finitely generated R-algebra, then

is a Jacobson ring. Further, if

is a maximal ideal, then

is a maximal ideal of R, and

is a finite extension field of

Another generalization states that a faithfully flat morphism of schemes

locally of finite type with X quasi-compact has a quasi-section, i.e. there exists

affine and faithfully flat and quasi-finite over X together with an X-morphism

Effective Nullstellensatz

In all of its variants, Hilbert's Nullstellensatz asserts that some polynomial {{mvar|g}} belongs or not to an ideal generated, say, by {{math|f₁, ..., f_k}}; we have {{math|g {{=}} f^r}} in the strong version, {{math|g {{=}} 1}} in the weak form. This means the existence or the non-existence of polynomials {{math|g₁, ..., g_k}} such that {{math|g {{=}} f₁g₁ + ... + f_kg_k}}. The usual proofs of the Nullstellensatz are not constructive, non-effective, in the sense that they do not give any way to compute the {{math|g_i}}.

It is thus a rather natural question to ask if there is an effective way to compute the {{math|g_i}} (and the exponent {{mvar|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 {{math|g_i}}: 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 {{math|g_i}}. 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 {{math|g_i}} 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.^[2] 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.

If {{mvar|d}} is the maximum of the degrees of the {{math|f_i}}, this bound may be simplified to

Kollár's result has been improved by several authors. {{As of|2012|10|14}}, the best improvement, due to M. Sombra is^[4]

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

Projective Nullstellensatz

We can formulate a certain correspondence between homogeneous ideals of polynomials and algebraic subsets of a projective space, called the projective Nullstellensatz, that is analogous to the affine one. To do that, we introduce some notations. Let

The homogeneous ideal,