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).

FormulationEdit

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,...,xn) 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 pr 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.

As a particular example, consider a point  . Then  . More generally,

 

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 generalizationEdit

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 also follows trivially from a systematic development of Jacobson rings, 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 NullstellensatzEdit

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.[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.

In the case of the weak Nullstellensatz, Kollár's bound is the following:[3]

Let f1, ..., fs be polynomials in n ≥ 2 variables, of total degree d1 ≥ ... ≥ ds. If there exist polynomials gi such that f1g1 + ... + fsgs = 1, then they can be chosen such that
 
This bound is optimal if all the degrees are greater than 2.

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[4]

 

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

Projective NullstellensatzEdit

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,

 

is called the maximal homogeneous ideal (see also irrelevant ideal). As in the affine case, we let: for a subset   and a homogeneous ideal I of R,

 

By   we mean: for every homogeneous coordinates   of a point of S we have  . This implies that the homogeneous components of f are also zero on S and thus that   is a homogeneous ideal. Equivalently,   is the homogeneous ideal generated by homogeneous polynomials f that vanish on S. Now, for any homogeneous ideal  , by the usual Nullstellensatz, we have:

 

and so, like in the affine case, we have:[5]

There exists an order-reversing one-to-one correspondence between proper homogeneous radical ideals of R and subsets of   of the form   The correspondence is given by   and  

Analytic NullstellensatzEdit

The Nullstellensatz also holds for the germs of holomorphic functions at a point of complex n-space   Precisely, for each open subset   let   denote the ring of holomorphic functions on U; then   is a sheaf on   The stalk   at, say, the origin can be shown to be a Noetherian local ring that is a unique factorization domain.

If   is a germ represented by a holomorphic function  , then let   be the equivalence class of the set

 

where two subsets   are considered equivalent if   for some neighborhood U of 0. Note   is independent of a choice of the representative   For each ideal   let   denote   for some generators   of I. It is well-defined; i.e., is independent of a choice of the generators.

For each subset  , let

 

It is easy to see that   is an ideal of   and that   if   in the sense discussed above.

The analytic Nullstellensatz then states:[6] for each ideal  ,

 

where the left-hand side is the radical of I.

See alsoEdit

NotesEdit

  1. ^ Atiyah–MacDonald 1969, Ch. 7
  2. ^ Brownawell, W. Dale (1987), "Bounds for the degrees in the Nullstellensatz", Ann. of Math., 126 (3): 577–591, doi:10.2307/1971361, MR 0916719
  3. ^ Kollár, János (1988), "Sharp Effective Nullstellensatz" (PDF), Journal of the American Mathematical Society, 1 (4): 963–975, doi:10.2307/1990996, MR 0944576, archived from the original (PDF) on 2014-03-03, retrieved 2012-10-14
  4. ^ Sombra, Martín (1999), "A Sparse Effective Nullstellensatz", Advances in Applied Mathematics, 22 (2): 271–295, arXiv:alg-geom/9710003, doi:10.1006/aama.1998.0633, MR 1659402
  5. ^ This formulation comes from Milne, Algebraic geometry [1] and differs from Hartshorne 1977, Ch. I, Exercise 2.4
  6. ^ Huybrechts, Proposition 1.1.29.

ReferencesEdit