Editing Hilbert's Nullstellensatz

{{short description|Relation between algebraic varieties and polynomial ideals}}
In [[mathematics]], '''Hilbert's Nullstellensatz''' (German for "theorem of zeros", or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between [[geometry]] and [[algebra]]. This relationship is the basis of [[algebraic geometry]]. It relates [[algebraic set]]s to [[ideal (ring theory)|ideals]] in [[polynomial ring]]s over [[algebraically closed field]]s. This relationship was discovered by [[David Hilbert]], 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]]).

== Formulation ==
Let <math>k</math> be a [[field (mathematics)|field]] (such as the [[rational number]]s) and <math>K</math> be an algebraically closed [[field extension]] of <math>k</math> (such as the [[complex number]]s). Consider the [[polynomial ring]] <math>k[X_1, \ldots, X_n]</math> and let <math>I</math> be an [[Ideal (ring theory)|ideal]] in this ring. The [[algebraic set]] <math>\mathrm V(I)</math> defined by this ideal consists of all <math>n</math>-tuples <math>\mathbf x = (x_1, \dots, x_n)</math> in <math>K^n</math> such that <math>f(\mathbf x) = 0</math> for all <math>f</math> in {{awrap|<math>I</math>.}} Hilbert's Nullstellensatz states that if ''p'' is some polynomial in <math>k[X_1, \ldots, X_n]</math> that vanishes on the algebraic set <math>\mathrm V(I)</math>, i.e. <math>p(\mathbf x) = 0</math> for all <math>\mathbf x</math> in <math>\mathrm V(I)</math>, then there exists a [[natural number]] <math>r</math> such that <math>p^r</math> is in <math>I</math>.<ref>{{harvnb|Zariski–Samuel|loc=Ch. VII, Theorem 14|last1=Zariski|year=1960}}.</ref>

An immediate corollary is the '''weak Nullstellensatz''': The ideal <math>I \subseteq k[X_1, \ldots, X_n]</math> contains 1 if and only if the polynomials in <math>I</math> do not have any common zeros in ''K<sup>n</sup>''. Specializing to the case <math>k=K=\mathbb{C}, n=1</math>, one immediately recovers a restatement of the [[fundamental theorem of algebra]]: a polynomial ''P'' in <math>\mathbb{C}[X]</math> has a root in <math>\mathbb{C}</math> 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.<ref>{{Cite book |last=Cox |first=David A. |url=https://link.springer.com/10.1007/978-3-319-16721-3 |title=Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra |last2=Little |first2=John |last3=O’Shea |first3=Donal |date=2015 |publisher=Springer International Publishing |isbn=978-3-319-16720-6 |series=Undergraduate Texts in Mathematics |location=Cham |language=en |doi=10.1007/978-3-319-16721-3}}</ref> The weak Nullstellensatz may also be formulated as follows: if ''I'' is a proper ideal in <math>k[X_1, \ldots, X_n],</math> then V(''I'') cannot be [[empty set|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''<sup>2</sup> + 1) in <math>\R[X]</math> do not have a common zero in <math>\R.</math>

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

:<math>\hbox{I}(\hbox{V}(J))=\sqrt{J}</math>

for every ideal ''J''. Here, <math>\sqrt{J}</math> denotes the [[radical of an ideal|radical]] of ''J'' and I(''U'') is the ideal of all polynomials that vanish on the set ''U''.

In this way, taking <math>k = K</math> we obtain an order-reversing [[bijective]] correspondence between the algebraic sets in ''K''<sup>''n''</sup> and the [[radical ideal]]s of <math>K[X_1, \ldots, X_n].</math> 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 operator]]s.

As a particular example, consider a point <math>P = (a_1, \dots, a_n) \in K^n</math>. Then <math>I(P) = (X_1 - a_1, \ldots, X_n - a_n)</math>. More generally,

:<math>\sqrt{I} = \bigcap_{(a_1, \dots, a_n) \in V(I)}  (X_1 - a_1, \dots, X_n - a_n).</math>

Conversely, every [[maximal ideal]] of the polynomial ring <math>K[X_1,\ldots,X_n]</math> (note that <math>K</math> is algebraically closed) is of the form <math>(X_1 - a_1, \ldots, X_n - a_n)</math> for some <math>a_1,\ldots,a_n \in K</math>.

As another example, an algebraic subset ''W'' in ''K''<sup>''n''</sup> is [[irreducible space|irreducible]] (in the Zariski topology) if and only if <math>I(W)</math> is a prime ideal.

== Proofs ==
There are many known proofs of the theorem. Some are [[constructive proof|non-constructive]], such as the first one. Others are constructive, as based on [[algorithm]]s for expressing {{math|1}} or {{math|''p{{sup|r}}''}} as a [[linear combination]] of the generators of the ideal.

===Using Zariski's lemma===
[[Zariski's lemma]] asserts that if a field is [[finitely generated algebra|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]]).  If <math>\mathfrak{m}</math> is a maximal ideal of <math>K[X_1,\ldots,X_n]</math> for algebraically closed ''K'', then Zariski's lemma implies that <math>K[X_1,\ldots,X_n]/ \mathfrak{m}</math> is a finite field extension of ''K'', and thus, by algebraic closure, must be ''K''.  From this, it follows that there is an <math>a = (a_1,\dots,a_n)\in K^n</math> such that <math>X_i-a_i\in\mathfrak{m}</math> for <math>i=1,\dots, n</math>.  In other words, 
:<math>\mathfrak{m} \supseteq \mathfrak{m}_a=(X_1 - a_1, \ldots, X_n - a_n)</math>
for some <math>a = (a_1,\dots,a_n)\in K^n</math>.  But <math>\mathfrak{m}_a</math> is clearly maximal, so <math>\mathfrak{m}=\mathfrak{m}_a</math>.  This is the weak Nullstellensatz: every maximal ideal of <math>K[X_1,\ldots,X_n]</math> for algebraically closed ''K'' is of the form <math>\mathfrak{m}_a=(X_1 - a_1, \ldots, X_n - a_n)</math> for some <math>a = (a_1,\dots,a_n)\in K^n</math>.  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.<ref>{{Cite book |last=Patil |first=Dilip P. |title=Introduction to Algebraic Geometry and Commutative Algebra |last2=Storch |first2=Uwe |publisher=World Scientific |year=2010 |isbn=978-9814307581}}</ref><ref>{{Cite book |last=Reid |first=Miles |title=Undergraduate commutative algebra |date=1995 |publisher=Cambridge University Press |isbn=978-0-521-45255-7 |series=London Mathematical Society student texts |location=Cambridge ; New York}}</ref> 

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.<ref>{{harvnb|Atiyah–Macdonald|loc=Ch. 7}}, Exercise 7.14.</ref>

Let <math>A = K[X_1, \ldots, X_n]</math> (''K'' an algebraically closed field), ''J'' an ideal of ''A,'' and <math>V=\mathrm{V}(J)</math> the common zeros of ''J'' in <math>K^n</math>. Clearly, <math>\sqrt{J} \subseteq \mathrm{I}(V)</math>, where <math>\mathrm{I}(V)</math> is the ideal of polynomials in ''A'' vanishing on ''V''. To show the opposite inclusion, let <math>f \not\in \sqrt{J}</math>. Then <math>f \not\in \mathfrak{p}</math> for some prime ideal <math>\mathfrak{p} \supseteq J</math> in ''A''. Let <math>R = (A/\mathfrak{p}) [1/\bar{f}]</math>, where <math>\bar{f}</math>is the image of ''f'' under the natural map <math>A \to A/\mathfrak{p}</math>, and <math>\mathfrak{m}</math> be a maximal ideal in ''R''. By Zariski's lemma, <math>R/\mathfrak{m}</math> is a finite extension of ''K'', and thus, is ''K'' since ''K'' is algebraically closed. Let <math>x_i</math> be the images of <math>X_i</math> under the natural map <math>A \to A/\mathfrak{p}\to R \to R/\mathfrak{m}\cong K</math>. It follows that, by construction, <math>x = (x_1, \ldots, x_n) \in V</math> but <math>f(x) \ne 0</math>, so <math>f \notin \mathrm{I}(V)</math>.

===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]] of two polynomials depending on a variable {{mvar|x}} 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 polynomial|monic]] in {{mvar|x}}, 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 ideal|principal]], generated by a non-constant polynomial {{mvar|p}} that depends on {{mvar|x}}, one chooses arbitrary values for the other variables. The [[fundamental theorem of algebra]] asserts that this choice can be extended to a zero of {{mvar|p}}.

In the case of several polynomials <math>p_1,\ldots, p_n,</math> a linear change of variables  allows to suppose that <math>p_1</math> is monic in the first variable {{mvar|x}}. Then, one introduces <math>n-1</math> new variables <math>u_2, \ldots, u_n,</math> and one considers the resultant
:<math>R=\operatorname{Res}_x(p_1,u_2p_2+\cdots +u_np_n).</math>

As {{mvar|R}} is in the ideal generated by <math>p_1,\ldots, p_n,</math> the same is true for  the coefficients in {{mvar|R}} of the [[monomial]]s in <math>u_2, \ldots, u_n.</math> So, if {{math|1}} is in the ideal generated by these coefficients, it is also in the ideal generated by <math>p_1,\ldots, p_n.</math> On the other hand, if these coefficients have a common zero, this zero can be extended to a common zero of <math>p_1,\ldots, p_n,</math> 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]] 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 {{math|1}} if and only if its [[reduced Gröbner basis]] (for any [[monomial ordering]]) is {{math|1}}.
*The number of the common zeros of the polynomials in a Gröbner basis is strongly related to the number of [[monomial]]s that are [[Gröbner basis#Reduction|irreducible]]s 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 order|lexicographic monomial order]], the common zeros can be computed by solving iteratively [[univariate polynomial]]s (this is not used in practice since one knows better algorithms).
* Strong Nullstellensatz: a power of {{mvar|p}} belongs to an ideal {{mvar|I}} if and only the [[Gröbner basis#Saturation|saturation]] of {{mvar|I}} by {{mvar|p}} produces the Gröbner basis {{math|1}}. 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 ring|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|''k''-algebra]] ''R'' (necessarily of the form <math display="inline">R = k[t_1,\cdots,t_n]/I</math>) is Jacobson. More generally, one has the following theorem:

: Let <math>R</math> be a Jacobson ring. If <math>S</math> is a finitely generated [[R-algebra|''R''-algebra]], then <math>S</math> is a Jacobson ring. Furthermore, if <math>\mathfrak{n}\subseteq S</math> is a maximal ideal, then <math>\mathfrak{m} := \mathfrak{n} \cap R</math> is a maximal ideal of <math display="inline">R</math>, and <math>S/\mathfrak{n}</math> is a finite extension of <math>R/\mathfrak{m}</math>.<ref>{{Cite web |last=Emerton |first=Matthew |title=Jacobson rings |url=http://www.math.uchicago.edu/~emerton/pdffiles/jacobson.pdf |url-status=live |archive-url=https://web.archive.org/web/20220725071929/http://www.math.uchicago.edu/~emerton/pdffiles/jacobson.pdf |archive-date=2022-07-25}}</ref>

Other generalizations proceed from viewing the Nullstellensatz in [[Scheme (mathematics)|scheme-theoretic]] terms as saying that for any field ''k'' and nonzero finitely generated ''k''-algebra ''R'', the morphism <math display="inline">\mathrm{Spec} \, R \to \mathrm{Spec} \, k</math> admits a [[Section (category theory)|section]] étale-locally (equivalently, after [[Base change (scheme theory)|base change]] along <math display="inline">\mathrm{Spec} \, L \to \mathrm{Spec} \, k</math> for some finite field extension <math display="inline">L/k</math>). In this vein, one has the following theorem:
:Any [[Faithfully flat morphism|faithfully flat]] morphism of schemes <math display="inline">f: Y \to X</math> [[Glossary of algebraic geometry#finite presentation|locally of finite presentation]] admits a ''quasi-section'', in the sense that there exists a faithfully flat and locally [[Quasi-finite morphism|quasi-finite]] morphism <math display="inline">g: X' \to X</math> locally of finite presentation such that the base change <math display="inline">f': Y \times_X X' \to X'</math> of <math display="inline">f</math> along <math display="inline">g</math> admits a section.<ref>[[Éléments de géométrie algébrique|EGA]] §IV.17.16.2.</ref> Moreover, if <math display="inline">X</math> is [[Quasi-compact morphism|quasi-compact]] (resp. quasi-compact and [[Quasi-separated morphism|quasi-separated]]), then one may take <math display="inline">X'</math> to be affine (resp. <math display="inline">X'</math> affine and <math display="inline">g</math> quasi-finite), and if <math display="inline">f</math> is [[Smooth morphism|smooth]] surjective, then one may take <math display="inline">g</math> to be [[Étale morphism|étale]].<ref>EGA §IV.17.16.3(ii).</ref>
[[Serge Lang]] gave an extension of the Nullstellensatz to the case of infinitely many generators:

:Let <math display="inline">\kappa</math> be an [[infinite cardinal]] and let <math display="inline">K</math> be an algebraically closed field whose [[transcendence degree]] over its [[prime subfield]] is strictly greater than <math>\kappa</math>. Then for any set <math display="inline">S</math> of cardinality <math display="inline">\kappa</math>, the polynomial ring <math display="inline">A = K[x_i]_{i \in S}</math> satisfies the Nullstellensatz, i.e., for any ideal <math display="inline">J \sub A</math> we have that <math>\sqrt{J} = \hbox{I} (\hbox{V} (J))</math>.<ref>{{Cite journal |last=Lang |first=Serge |date=1952 |title=Hilbert's Nullstellensatz in Infinite-Dimensional Space |url=https://www.jstor.org/stable/2031893 |journal=[[Proc. Am. Math. Soc.]] |volume=3 |issue=3 |pages=407–410 |doi=10.2307/2031893 |jstor=2031893 |via=}}</ref>

== 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|''f''<sub>1</sub>, ..., ''f<sub>k</sub>''}}; we have {{math|''g'' {{=}} ''f<sup>&thinsp;r</sup>''}} in the strong version, {{math|''g'' {{=}} 1}} in the weak form. This means the existence or the non-existence of polynomials {{math|''g''<sub>1</sub>, ..., ''g<sub>k</sub>''}} such that {{math|''g'' {{=}} ''f''<sub>1</sub>''g''<sub>1</sub> + ... + ''f<sub>k</sub>g<sub>k</sub>''}}. 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<sub>i</sub>''}}.

It is thus a rather natural question to ask if there is an effective way to compute the {{math|''g<sub>i</sub>''}} (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<sub>i</sub>''}}: 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<sub>i</sub>''}}. 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<sub>i</sub>''}} 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.<ref>{{citation|last=Brownawell|first=W. Dale|title=Bounds for the degrees in the Nullstellensatz|journal=[[Ann. of Math.]]|volume=126|year=1987|issue=3|pages=577–591|mr=0916719|doi=10.2307/1971361|jstor=1971361 }}</ref>  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:<ref>{{citation| first=János| last=Kollár| title=Sharp Effective Nullstellensatz| journal=[[Journal of the American Mathematical Society]]| volume=1| issue=4| year=1988| pages=963–975| url=http://www.math.ucdavis.edu/~deloera/MISC/BIBLIOTECA/trunk/Kollar/kollarnullstellen.pdf| doi=10.2307/1990996| jstor=1990996| mr=0944576| access-date=2012-10-14| archive-url=https://web.archive.org/web/20140303174253/https://www.math.ucdavis.edu/~deloera/MISC/BIBLIOTECA/trunk/Kollar/kollarnullstellen.pdf| archive-date=2014-03-03| url-status=dead}}</ref>

:Let {{math|''f''<sub>1</sub>, ..., ''f<sub>s</sub>''}} be polynomials in {{math|''n'' ≥ 2}} variables, of total degree {{math|''d''<sub>1</sub> ≥ ... ≥ ''d<sub>s</sub>''}}. If there exist polynomials {{math|''g<sub>i</sub>''}} such that {{math|''f''<sub>1</sub>''g''<sub>1</sub> + ... + ''f<sub>s</sub>g<sub>s</sub>'' {{=}} 1}}, then they can be chosen such that 
::<math>\deg(f_ig_i) \le \max(d_s,3)\prod_{j=1}^{\min(n,s)-1}\max(d_j,3).</math> 
:This bound is optimal if all the degrees are greater than 2.

If {{mvar|d}} is the maximum of the degrees of the {{math|''f<sub>i</sub>''}}, this bound may be simplified to 
:<math>\max(3,d)^{\min(n,s)}.</math>

An improvement due to M. Sombra is<ref>{{citation|first=Martín|last=Sombra|title=A Sparse Effective Nullstellensatz| journal=[[Advances in Applied Mathematics]]|volume= 22|issue=2|year=1999| pages=271–295 |arxiv=alg-geom/9710003|doi=10.1006/aama.1998.0633 | mr=1659402|s2cid=119726673 }}</ref>

:<math>\deg(f_ig_i) \le 2d_s\prod_{j=1}^{\min(n,s)-1}d_j.</math>

His bound improves Kollár's as soon as at least two of the degrees that are involved are lower than&nbsp;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 <math>R = k[t_0, \ldots, t_n].</math> The homogeneous ideal, 

:<math>R_+ = \bigoplus_{d \geqslant 1} R_d</math> 

is called the ''maximal homogeneous ideal'' (see also [[irrelevant ideal]]). As in the affine case, we let: for a subset <math>S \subseteq \mathbb{P}^n</math> and a homogeneous ideal ''I'' of ''R'',

:<math>\begin{align}
\operatorname{I}_{\mathbb{P}^n}(S) &= \{ f \in R_+ \mid f = 0 \text{ on } S \}, \\
\operatorname{V}_{\mathbb{P}^n}(I) &= \{ x \in \mathbb{P}^n \mid f(x) = 0 \text{ for all } f \in I \}.
\end{align}</math>

By <math>f = 0 \text{ on } S</math> we mean: for every homogeneous coordinates <math>(a_0 : \cdots : a_n)</math> of a point of ''S'' we have <math>f(a_0,\ldots, a_n)=0</math>. This implies that the homogeneous components of ''f'' are also zero on ''S'' and thus that <math>\operatorname{I}_{\mathbb{P}^n}(S)</math> is a homogeneous ideal. Equivalently, <math>\operatorname{I}_{\mathbb{P}^n}(S)</math> is the homogeneous ideal generated by homogeneous polynomials ''f'' that vanish on ''S''. Now, for any homogeneous ideal <math>I \subseteq R_+</math>, by the usual Nullstellensatz, we have:

:<math>\sqrt{I} = \operatorname{I}_{\mathbb{P}^n}(\operatorname{V}_{\mathbb{P}^n}(I)),</math>

and so, like in the affine case, we have:<ref>This formulation comes from Milne, Algebraic geometry [http://www.jmilne.org/math/CourseNotes/ag.html] and differs from {{harvnb|Hartshorne|1977|loc=Ch. I, Exercise 2.4}}</ref>

:There exists an order-reversing one-to-one correspondence between proper homogeneous radical ideals of ''R'' and subsets of <math>\mathbb{P}^n</math> of the form <math>\operatorname{V}_{\mathbb{P}^n}(I).</math> The correspondence is given by <math>\operatorname{I}_{\mathbb{P}^n}</math> and <math>\operatorname{V}_{\mathbb{P}^n}.</math>

== Analytic Nullstellensatz (Rückert’s Nullstellensatz)==
The Nullstellensatz also holds for the germs of holomorphic functions at a point of complex ''n''-space <math>\Complex^n.</math> Precisely, for each open subset <math>U \subseteq \Complex^n,</math> let <math>\mathcal{O}_{\Complex^n}(U)</math> denote the ring of holomorphic functions on ''U''; then <math>\mathcal{O}_{\Complex^n}</math> is a ''[[sheaf (mathematics)|sheaf]]'' on <math>\Complex^n.</math> The stalk <math>\mathcal{O}_{\Complex^n, 0}</math> at, say, the origin can be shown to be a [[Noetherian ring|Noetherian]] [[local ring]] that is a [[unique factorization domain]].

If <math>f \in \mathcal{O}_{\Complex ^n, 0}</math> is a germ represented by a holomorphic function <math>\widetilde{f}: U \to \Complex </math>, then let <math>V_0(f)</math> be the equivalence class of the set 

:<math>\left \{ z \in U \mid \widetilde{f}(z) = 0  \right\},</math>

where two subsets <math>X, Y \subseteq \Complex^n</math> are considered equivalent if <math>X \cap U = Y \cap U</math> for some neighborhood ''U'' of 0. Note <math>V_0(f)</math> is independent of a choice of the representative <math>\widetilde{f}.</math> For each ideal <math>I \subseteq \mathcal{O}_{\Complex^n,0},</math> let <math>V_0(I)</math> denote <math>V_0(f_1) \cap \dots \cap V_0(f_r)</math> for some generators <math>f_1, \ldots, f_r</math> of ''I''. It is well-defined; i.e., is independent of a choice of the generators.

For each subset <math>X \subseteq \Complex ^n</math>, let 

:<math>I_0(X) = \left \{ f \in \mathcal{O}_{\Complex^n,0} \mid V_0(f) \supset X \right \}.</math>

It is easy to see that <math>I_0(X)</math> is an ideal of <math>\mathcal{O}_{\Complex ^n, 0}</math> and that <math>I_0(X) = I_0(Y)</math> if <math>X \sim Y</math> in the sense discussed above.

The '''analytic Nullstellensatz''' then states:<ref>{{harvnb|Huybrechts|loc=Proposition 1.1.29.}}</ref> for each ideal <math>I \subseteq \mathcal{O}_{\Complex ^n, 0}</math>,

:<math>\sqrt{I} = I_0(V_0(I))</math>

where the left-hand side is the [[radical of an ideal|radical]] of ''I''.<!-- TODO: need to mention an analytic subset, which apparently is yet to be created. -->

==See also==
*[[Artin–Tate lemma]]
*[[Combinatorial Nullstellensatz]]
*[[Differential Nullstellensatz]]
*[[Real radical]]<!-- the article should probably have a section on the real situation, which is more complex!. -->
*[[Restricted power series#Tate algebra]], an analog of Hilbert's Nullstellensatz holds for Tate algebras.
*[[Stengle's Positivstellensatz]]
*[[Weierstrass Nullstellensatz]]

==Notes==
{{reflist}}

==References==
* {{Cite journal |last=Almira |first=Jose María |date=2007 |title=Nullstellensatz revisited |url=http://www.seminariomatematico.polito.it/rendiconti/65-3/365.pdf |journal=[[Rend. Semin. Mat. Univ. Politec. Torino]] |volume=65 |issue=3 |pages=365–369}}
*{{Cite book|ref=CITEREFAtiyah–Macdonald|last1=Atiyah |first1=M.F. |title=Introduction to Commutative Algebra |title-link=Introduction to Commutative Algebra |last2=Macdonald |first2=I.G. |publisher=Addison-Wesley |year=1994 |isbn=0-201-40751-5 |author-link=Michael Atiyah |author-link2=Ian G. Macdonald}}
* {{Cite book |last=Eisenbud |first=David |title=Commutative Algebra With a View Toward Algebraic Geometry |publisher=Springer-Verlag |year=1999 |isbn=978-0-387-94268-1 |series=Graduate Texts in Mathematics |volume=150 |location=New York |author-link=David Eisenbud}}
*{{Hartshorne AG}}
*{{Cite journal |last=Hilbert |first=David |author-link=David Hilbert |date=1893 |title=Ueber die vollen Invariantensysteme |journal=[[Mathematische Annalen]] |volume=42 |issue=3 |pages=313–373 |doi=10.1007/BF01444162 | url=https://resolver.sub.uni-goettingen.de/purl?PPN235181684_0042 }}
*{{cite book|ref=CITEREFHuybrechts|title=Complex Geometry: An Introduction|first=Daniel|last=Huybrechts |author-link=Daniel Huybrechts
|publisher=Springer|year=2005|isbn=3-540-21290-6}}
*{{cite book |author=Mukai |first=Shigeru |authorlink=Shigeru Mukai |others=William Oxbury (trans.) |title=An Introduction to Invariants and Moduli |series=Cambridge studies in advanced mathematics |volume=81 |year=2003 |isbn=0-521-80906-1 |page=82}}
* {{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}}


[[Category:Polynomials]]
[[Category:Theorems in algebraic geometry]]