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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Hilbert's Nullstellensatz (German for "theorem of zeros", or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
and
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
. This relationship is the basis of
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
. It relates algebraic sets to ideals in
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
s over
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...
s. This relationship was discovered by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...
, 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 In mathematics Hilbert's basis theorem asserts that every ideal (ring theory), ideal of a polynomial ring over a field (mathematics), field has a finite generating set of an ideal, generating set (a finite ''basis'' in Hilbert's terminology). In ...
).


Formulation

Let k be a field (such as the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s) and K be an algebraically closed
field extension In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...
of k (such as the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s). Consider the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
k _1, \ldots, X_n/math> and let I be an ideal in this ring. The algebraic set \mathrm V(I) defined by this ideal consists of all n-tuples \mathbf x = (x_1, \dots, x_n) in K^n such that f(\mathbf x) = 0 for all f in Hilbert's Nullstellensatz states that if ''p'' is some polynomial in k _1, \ldots, X_n/math> that vanishes on the algebraic set \mathrm V(I), i.e. p(\mathbf x) = 0 for all \mathbf x in \mathrm V(I), then there exists a
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
r such that p^r is in I. An immediate corollary is the weak Nullstellensatz: The ideal I \subseteq k _1, \ldots, X_n/math> contains 1 if and only if the polynomials in I do not have any common zeros in ''Kn''. Specializing to the case k=K=\mathbb, n=1, one immediately recovers a restatement of the fundamental theorem of algebra: a polynomial ''P'' in \mathbb /math> has a root in \mathbb if and only if deg ''P'' ≠ 0. For this reason, the (weak) Nullstellensatz has been referred to as a generalization of the fundamental theorem of algebra for multivariable polynomials. The weak Nullstellensatz may also be formulated as follows: if ''I'' is a proper ideal in k _1, \ldots, X_n 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 (''X''2 + 1) in \R /math> do not have a common zero in \R. With the notation common in algebraic geometry, the Nullstellensatz can also be formulated as :\hbox(\hbox(J))=\sqrt for every ideal ''J''. Here, \sqrt denotes the radical of ''J'' and I(''U'') is the ideal of all polynomials that vanish on the set ''U''. In this way, taking k = K we obtain an order-reversing
bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
correspondence between the algebraic sets in ''K''''n'' and the radical ideals of K _1, \ldots, X_n In fact, more generally, one has a
Galois connection In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fun ...
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 P = (a_1, \dots, a_n) \in K^n. Then I(P) = (X_1 - a_1, \ldots, X_n - a_n). More generally, :\sqrt = \bigcap_ (X_1 - a_1, \dots, X_n - a_n). Conversely, every
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
of the polynomial ring K _1,\ldots,X_n/math> (note that K is algebraically closed) is of the form (X_1 - a_1, \ldots, X_n - a_n) for some a_1,\ldots,a_n \in K. As another example, an algebraic subset ''W'' in ''K''''n'' is irreducible (in the Zariski topology) if and only if I(W) is a prime ideal.


Proofs

There are many known proofs of the theorem. Some are non-constructive, such as the first one. Others are constructive, as based on
algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...
s for expressing or as a
linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of the generators of the ideal.


Using Zariski's lemma

Zariski's lemma asserts that if a field is finitely generated as an
associative algebra In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...
over a field ''K'', then it is a finite field extension of ''K'' (that is, it is also finitely generated as a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
). If ''K'' is an algebraically closed field and \mathfrak is a maximal ideal of the ring of polynomials K _1,\ldots,X_n/math>, then Zariski's lemma implies that K _1,\ldots,X_n \mathfrak is a finite field extension of ''K'', and thus, by algebraic closure, must be ''K''. From this, it follows that there is an a = (a_1,\dots,a_n)\in K^n such that X_i-a_i\in\mathfrak for i=1,\dots, n. In other words, :\mathfrak \supseteq \mathfrak_a=(X_1 - a_1, \ldots, X_n - a_n) for some a = (a_1,\dots,a_n)\in K^n. But \mathfrak_a is clearly maximal, so \mathfrak=\mathfrak_a. This is the weak Nullstellensatz: every maximal ideal of K _1,\ldots,X_n/math> for algebraically closed ''K'' is of the form \mathfrak_a=(X_1 - a_1, \ldots, X_n - a_n) for some a = (a_1,\dots,a_n)\in K^n. Because of this close relationship, some texts refer to Zariski's lemma as the weak Nullstellensatz or as the 'algebraic version' of the weak Nullstellensatz. The full Nullstellensatz can also be proved directly from Zariski's lemma without employing the Rabinowitsch trick. Here is a sketch of a proof using this lemma. Let A = K _1, \ldots, X_n/math> for algebraically closed field ''K'', and let ''J'' be an ideal of ''A'' and V=\mathrm(J) be the common zeros of ''J'' in K^n. Clearly, \sqrt \subseteq \mathrm(V), where \mathrm(V) is the ideal of polynomials in ''A'' vanishing on ''V''. To show the opposite inclusion, let f \not\in \sqrt. Then f \not\in \mathfrak for some prime ideal \mathfrak \supseteq J in ''A''. Let R = (A/\mathfrak) /\bar/math>, where \baris the image of ''f'' under the natural map A \to A/\mathfrak, and \mathfrak be a maximal ideal in ''R''. By Zariski's lemma, R/\mathfrak is a finite extension of ''K'', and thus, is ''K'' since ''K'' is algebraically closed. Let x_i be the images of X_i under the natural map A \to A/\mathfrak\to R \to R/\mathfrak\cong K. It follows that, by construction, x = (x_1, \ldots, x_n) \in V but f(x) \ne 0, so f \notin \mathrm(V).


Using resultants

The following constructive proof of the weak form is one of the oldest proofs (the strong form results from the Rabinowitsch trick, which is also constructive). The
resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over th ...
of two polynomials depending on a variable and other variables is a polynomial in the other variables that is in the ideal generated by the two polynomials, and has the following properties: if one of the polynomials is monic in , every zero (in the other variables) of the resultant may be extended into a common zero of the two polynomials. The proof is as follows. If the ideal is principal, generated by a non-constant polynomial that depends on , one chooses arbitrary values for the other variables. The fundamental theorem of algebra asserts that this choice can be extended to a zero of . In the case of several polynomials p_1,\ldots, p_n, a linear change of variables allows to suppose that p_1 is monic in the first variable . Then, one introduces n-1 new variables u_2, \ldots, u_n, and one considers the resultant :R=\operatorname_x(p_1,u_2p_2+\cdots +u_np_n). As is in the ideal generated by p_1,\ldots, p_n, the same is true for the coefficients in of the
monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called a power product or primitive monomial, is a product of powers of variables with n ...
s in u_2, \ldots, u_n. So, if is in the ideal generated by these coefficients, it is also in the ideal generated by p_1,\ldots, p_n. On the other hand, if these coefficients have a common zero, this zero can be extended to a common zero of p_1,\ldots, p_n, by the above property of the resultant. This proves the weak Nullstellensatz by induction on the number of variables.


Using Gröbner bases

A
Gröbner basis In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring K _1,\ldots,x_n/math> ove ...
is an algorithmic concept that was introduced in 1973 by Bruno Buchberger. It is presently fundamental in computational geometry. A Gröbner basis is a special generating set of an ideal from which most properties of the ideal can easily be extracted. Those that are related to the Nullstellensatz are the following: *An ideal contains if and only if its reduced Gröbner basis (for any
monomial ordering In mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all (Monic polynomial, monic) monomials in a given polynomial ring, satisfying the property of respecting multiplication, i.e., ...
) is . *The number of the common zeros of the polynomials in a Gröbner basis is strongly related to the number of
monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called a power product or primitive monomial, is a product of powers of variables with n ...
s that are irreducibles by the basis. Namely, the number of common zeros is infinite if and only if the same is true for the irreducible monomials; if the two numbers are finite, the number of irreducible monomials equals the numbers of zeros (in an algebraically closed field), counted with multiplicities. *With a lexicographic monomial order, the common zeros can be computed by solving iteratively
univariate polynomial In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer ...
s (this is not used in practice since one knows better algorithms). * Strong Nullstellensatz: a power of belongs to an ideal if and only the saturation of by produces the Gröbner basis . Thus, the strong Nullstellensatz results almost immediately from the definition of the saturation.


Generalizations

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 R = k _1,\cdots,t_nI) is Jacobson. More generally, one has the following theorem: : Let R be a Jacobson ring. If S is a finitely generated ''R''-algebra, then S is a Jacobson ring. Furthermore, if \mathfrak\subseteq S is a maximal ideal, then \mathfrak := \mathfrak \cap R is a maximal ideal of R, and S/\mathfrak is a finite extension of R/\mathfrak. Other generalizations proceed from viewing the Nullstellensatz in scheme-theoretic terms as saying that for any field ''k'' and nonzero finitely generated ''k''-algebra ''R'', the morphism \mathrm \, R \to \mathrm \, k admits a section étale-locally (equivalently, after base change along \mathrm \, L \to \mathrm \, k for some finite field extension L/k). In this vein, one has the following theorem: :Any faithfully flat morphism of schemes f: Y \to X locally of finite presentation admits a ''quasi-section'', in the sense that there exists a faithfully flat and locally quasi-finite morphism g: X' \to X locally of finite presentation such that the base change f': Y \times_X X' \to X' of f along g admits a section. Moreover, if X is quasi-compact (resp. quasi-compact and quasi-separated), then one may take X' to be affine (resp. X' affine and g quasi-finite), and if f is smooth surjective, then one may take g to be étale. Serge Lang gave an extension of the Nullstellensatz to the case of infinitely many generators: :Let \kappa be an infinite cardinal and let K be an algebraically closed field whose transcendence degree over its prime subfield is strictly greater than \kappa. Then for any set S of cardinality \kappa, the polynomial ring A = K _i satisfies the Nullstellensatz, i.e., for any ideal J \sub A we have that \sqrt = \hbox (\hbox (J)).


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 In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variable (math), variables. For example, : \begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of th ...
that may be solved by usual
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...
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: :Let be polynomials in variables, of total degree . If there exist polynomials such that , then they can be chosen such that ::\deg(f_ig_i) \le \max(d_s,3)\prod_^\max(d_j,3). :This bound is optimal if all the degrees are greater than 2. If is the maximum of the degrees of the , this bound may be simplified to :\max(3,d)^. An improvement due to M. Sombra is :\deg(f_ig_i) \le 2d_s\prod_^d_j. 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 R = k _0, \ldots, t_n The homogeneous ideal, :R_+ = \bigoplus_ R_d is called the ''maximal homogeneous ideal'' (see also irrelevant ideal). As in the affine case, we let: for a subset S \subseteq \mathbb^n and a homogeneous ideal ''I'' of ''R'', :\begin \operatorname_(S) &= \, \\ \operatorname_(I) &= \. \end By f = 0 \text S we mean: for every homogeneous coordinates (a_0 : \cdots : a_n) of a point of ''S'' we have f(a_0,\ldots, a_n)=0. This implies that the homogeneous components of ''f'' are also zero on ''S'' and thus that \operatorname_(S) is a homogeneous ideal. Equivalently, \operatorname_(S) is the homogeneous ideal generated by homogeneous polynomials ''f'' that vanish on ''S''. Now, for any homogeneous ideal I \subseteq R_+, by the usual Nullstellensatz, we have: :\sqrt = \operatorname_(\operatorname_(I)), and so, like in the affine case, we have: :There exists an order-reversing one-to-one correspondence between proper homogeneous radical ideals of ''R'' and subsets of \mathbb^n of the form \operatorname_(I). The correspondence is given by \operatorname_ and \operatorname_.


Analytic Nullstellensatz (Rückert’s Nullstellensatz)

The Nullstellensatz also holds for the germs of holomorphic functions at a point of complex ''n''-space \Complex^n. Precisely, for each open subset U \subseteq \Complex^n, let \mathcal_(U) denote the ring of holomorphic functions on ''U''; then \mathcal_ is a '' sheaf'' on \Complex^n. The stalk \mathcal_ at, say, the origin can be shown to be a
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
local ring In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of ...
that is a
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
. If f \in \mathcal_ is a germ represented by a holomorphic function \widetilde: U \to \Complex , then let V_0(f) be the equivalence class of the set :\left \, where two subsets X, Y \subseteq \Complex^n are considered equivalent if X \cap U = Y \cap U for some neighborhood ''U'' of 0. Note V_0(f) is independent of a choice of the representative \widetilde. For each ideal I \subseteq \mathcal_, let V_0(I) denote V_0(f_1) \cap \dots \cap V_0(f_r) for some generators f_1, \ldots, f_r of ''I''. It is well-defined; i.e., is independent of a choice of the generators. For each subset X \subseteq \Complex ^n, let :I_0(X) = \left \. It is easy to see that I_0(X) is an ideal of \mathcal_ and that I_0(X) = I_0(Y) if X \sim Y in the sense discussed above. The analytic Nullstellensatz then states: for each ideal I \subseteq \mathcal_, :\sqrt = I_0(V_0(I)) where the left-hand side is the radical of ''I''.


See also

* Artin–Tate lemma * Combinatorial Nullstellensatz * Differential Nullstellensatz * Real radical * Restricted power series#Tate algebra, an analog of Hilbert's Nullstellensatz holds for Tate algebras. * Stengle's Positivstellensatz * Weierstrass Nullstellensatz


Notes


References

* * * * * * * * {{cite book , ref=CITEREFZariski–Samuel, last1=Zariski , first1=Oscar , last2=Samuel , first2=Pierre , author1-link=Oscar Zariski , author2-link=Pierre Samuel , title=Commutative algebra. Volume II , date=1960 , location=Berlin , isbn=978-3-662-27753-9 Polynomials Theorems in algebraic geometry