Hilbert's Nullstellensatz
In mathematics, 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. 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 formula_1 and let "I" be an ideal in this ring. The algebraic set V("I") defined by this ideal consists of all "n"-tuples x = ("x"...,"x") in "K" such that "f"(x) = 0 for all "f" in "I". Hilbert's Nullstellensatz states that if "p" is some polynomial in formula_1 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" is in "I". An immediate corollary is the "weak Nullstellensatz": The ideal formula_3 contains 1 if and only  if the polynomials in "I" do not have any common zeros in "K". It may also be formulated as follows: if "I" is a proper ideal in formula_4 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 ("X  + 1) in formula_5 do not have a common zero in formula_6 With the notation common in algebraic geometry, the Nullstellensatz can also be formulated as for every ideal J". Here, formula_8 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 "K" and the radical ideals of formula_9 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 formula_10. Then formula_11. More generally, Conversely, every maximal ideal of the polynomial ring formula_13 (note that formula_14 is algebraically closed) is of the form formula_15 for some formula_16. As another example, an algebraic subset "W" in "K" is irreducible (in the Zariski topology) if and only if formula_17 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. Let formula_18 ("k" algebraically closed field), "I" an ideal of "A," and "V" the common zeros of "I" in formula_19. Clearly, formula_20. Let formula_21. Then formula_22 for some prime ideal formula_23 in "A". Let formula_24 and formula_25 a maximal ideal in formula_26. By Zariski's lemma, formula_27 is a finite extension of "k"; thus, is "k" since "k" is algebraically closed. Let formula_28 be the images of formula_29 under the natural map formula_30 passing through  formula_26. It follows that formula_32 and formula_33. The Nullstellensatz also follows trivially from a systematic development of Jacobson rings, in which a radical ideal is an intersection of maximal ideals. Let formula_26 be a Jacobson ring. If formula_35 is a finitely generated "R"-algebra, then formula_35 is a Jacobson ring. Further, if formula_37 is a maximal ideal, then formula_38 is a maximal ideal of R, and formula_39 is a finite extension field of formula_27. Another generalization states that a faithfully flat morphism of schemes formula_41 locally of finite type with "X" quasi-compact has a "quasi-section", i.e. there exists formula_42  affine and faithfully flat and quasi-finite over "X" together with an "X"-morphism formula_43 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 (and the exponent 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 : 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 Kollár's result has been improved by several authors. , the best 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. 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 formula_47 The homogeneous ideal, is called the "maximal homogeneous ideal" (see also irrelevant ideal). As in the affine case, we let: for a subset formula_49 and a homogeneous ideal "I" of "R", By formula_51 we mean : for every homogeneous coordinates formula_52 of a point of "S" we have formula_53. This implies that the homogeneous components of "f" are also zero on "S" and thus that formula_54 is a homogeneous ideal. Equivalently, formula_54 is the homogeneous ideal generated by homogeneous polynomials "f" that vanish on "S". Now, for any homogeneous ideal formula_56, by the usual Nullstellensatz, we have: and so, like in the affine case, we have: Analytic Nullstellensatz (Rückert’s Nullstellensatz). The Nullstellensatz also holds for the germs of holomorphic functions at a point of complex "n"-space formula_62 Precisely, for each open subset formula_63 let formula_64  denote the ring of holomorphic functions on "U"; then formula_65 is a "sheaf" on formula_62 The stalk formula_67 at, say, the origin can be shown to be a Noetherian local ring that is a unique factorization domain. If formula_68 is a germ represented by a holomorphic function formula_69, then let formula_70 be the equivalence class of the set where two subsets formula_72 are considered equivalent if formula_73 for some neighborhood "U" of 0. Note formula_70 is independent of a choice of the representative formula_75 For each ideal formula_76 let formula_77 denote formula_78 for some generators formula_79 of "I". It is  well-defined; i.e., is independent of a choice of the generators. For each subset formula_80, let It is easy to see that formula_82 is an ideal of formula_83 and that formula_84 if formula_85 in the sense discussed above. The analytic Nullstellensatz then states: for each ideal formula_86, where the left-hand side is the radical of "I".