Inmathematics,Hilbert's Nullstellensatz (German for "theorem of zeros", or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship betweengeometry andalgebra. This relationship is the basis ofalgebraic geometry. It relatesalgebraic sets toideals inpolynomial rings overalgebraically closed fields. This relationship was discovered byDavid Hilbert, who proved the Nullstellensatz in his second major paper oninvariant theory in 1893 (following his seminal 1890 paper in which he provedHilbert's basis theorem).

Let${\displaystyle k}$ be afield (such as therational numbers) and${\displaystyle K}$ be an algebraically closedfield extension of${\displaystyle k}$ (such as thecomplex numbers). Consider thepolynomial ring${\displaystyle k[X_1,\dots ,X_n]}$ and let${\displaystyle I}$ be anideal in this ring. Thealgebraic set${\displaystyle \mathrm{V}(I)}$ defined by this ideal consists of all${\displaystyle n}$-tuples${\displaystyle \mathbf{x}=(x_1,\dots ,x_n)}$ in${\displaystyle K^n}$ such that${\displaystyle f(\mathbf{x})=0}$ for all${\displaystyle f}$ in${\displaystyle I}$. Hilbert's Nullstellensatz states that ifp is some polynomial in${\displaystyle k[X_1,\dots ,X_n]}$ that vanishes on the algebraic set${\displaystyle \mathrm{V}(I)}$, i.e.${\displaystyle p(\mathbf{x})=0}$ for all${\displaystyle \mathbf{x}}$ in${\displaystyle \mathrm{V}(I)}$, then there exists anatural number${\displaystyle r}$ such that${\displaystyle p^r}$ is in${\displaystyle I}$.^[1]

An immediate corollary is theweak Nullstellensatz: The ideal${\displaystyle I\subseteq k[X_1,\dots ,X_n]}$ contains 1 if and only if the polynomials in${\displaystyle I}$ do not have any common zeros inKⁿ. Specializing to the case${\displaystyle k=K=\mathbb{C},n=1}$, one immediately recovers a restatement of thefundamental theorem of algebra: a polynomialP in${\displaystyle \mathbb{C}[X]}$ has a root in${\displaystyle \mathbb{C}}$ if and only if degP ≠ 0. For this reason, the (weak) Nullstellensatz has been referred to as a generalization of the fundamental theorem of algebra for multivariable polynomials.^[2] The weak Nullstellensatz may also be formulated as follows: ifI is a proper ideal in${\displaystyle k[X_1,\dots ,X_n],}$ then V(I) cannot beempty, i.e. there exists a common zero for all the polynomials in the ideal in every algebraically closed extension ofk. This is the reason for the name of the theorem, the full version of which can be proved easily from the 'weak' form using theRabinowitsch 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${\displaystyle \mathbb{R}[X]}$ do not have a common zero in${\displaystyle \mathbb{R}.}$

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

${\displaystyle \text{I}(\text{V}(J))=\sqrt{J}}$

for every idealJ. Here,${\displaystyle \sqrt{J}}$ denotes theradical ofJ and I(U) is the ideal of all polynomials that vanish on the setU.

In this way, taking${\displaystyle k=K}$ we obtain an order-reversingbijective correspondence between the algebraic sets inKⁿ and theradical ideals of${\displaystyle K[X_1,\dots ,X_n].}$ In fact, more generally, one has aGalois connection between subsets of the space and subsets of the algebra, where "Zariski closure" and "radical of the ideal generated" are theclosure operators.

As a particular example, consider a point${\displaystyle P=(a_1,\dots ,a_n)\in K^n}$. Then${\displaystyle I(P)=(X_1-a_1,\dots ,X_n-a_n)}$. More generally,

${\displaystyle \sqrt{I}=\bigcap _{(a_1,\dots ,a_n)\in V(I)}(X_1-a_1,\dots ,X_n-a_n).}$

Conversely, everymaximal ideal of the polynomial ring${\displaystyle K[X_1,\dots ,X_n]}$ (note that${\displaystyle K}$ is algebraically closed) is of the form${\displaystyle (X_1-a_1,\dots ,X_n-a_n)}$ for some${\displaystyle a_1,\dots ,a_n\in K}$.

As another example, an algebraic subsetW inKⁿ isirreducible (in the Zariski topology) if and only if${\displaystyle I(W)}$ is a prime ideal.

There are many known proofs of the theorem. Some arenon-constructive, such as the first one. Others are constructive, as based onalgorithms for expressing1 orp^r as alinear combination of the generators of the ideal.

Zariski's lemma asserts that if a field isfinitely generated as anassociative algebra over a fieldk, then it is afinite field extension ofk (that is, it is also finitely generated as avector space).

Here is a sketch of a proof using this lemma.^[3]

Let${\displaystyle A=k[t_1,\dots ,t_n]}$ (k algebraically closed field),I an ideal ofA, andV the common zeros ofI in${\displaystyle k^n}$. Clearly,${\displaystyle \sqrt{I}\subseteq I(V)}$. Let${\displaystyle f\notin \sqrt{I}}$. Then${\displaystyle f\notin \mathfrak{p}}$ for some prime ideal${\displaystyle \mathfrak{p}\supseteq I}$ inA. Let${\displaystyle R=(A/\mathfrak{p})[f^{-1}]}$ and${\displaystyle \mathfrak{m}}$ a maximal ideal in${\displaystyle R}$. By Zariski's lemma,${\displaystyle R/\mathfrak{m}}$ is a finite extension ofk; thus, isk sincek is algebraically closed. Let${\displaystyle x_i}$ be the images of${\displaystyle t_i}$ under the natural map${\displaystyle A\to k}$ passing through${\displaystyle R}$. It follows that${\displaystyle x=(x_1,\dots ,x_n)\in V}$ and${\displaystyle f(x)\ne 0}$.

The following constructive proof of the weak form is one of the oldest proofs (the strong form results from theRabinowitsch trick, which is also constructive).

Theresultant of two polynomials depending on a variablex 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 ismonic inx, 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 isprincipal, generated by a non-constant polynomialp that depends onx, one chooses arbitrary values for the other variables. Thefundamental theorem of algebra asserts that this choice can be extended to a zero ofp.

In the case of several polynomials${\displaystyle p_1,\dots ,p_n,}$ a linear change of variables allows to suppose that${\displaystyle p_1}$ is monic in the first variablex. Then, one introduces${\displaystyle n-1}$ new variables${\displaystyle u_2,\dots ,u_n,}$ and one considers the resultant

${\displaystyle R={\mathrm{Res}}_x(p_1,u_2{p}_2+\cdots +u_n{p}_n).}$

AsR is in the ideal generated by${\displaystyle p_1,\dots ,p_n,}$ the same is true for the coefficients inR of themonomials in${\displaystyle u_2,\dots ,u_n.}$ So, if1 is in the ideal generated by these coefficients, it is also in the ideal generated by${\displaystyle p_1,\dots ,p_n.}$ On the other hand, if these coefficients have a common zero, this zero can be extended to a common zero of${\displaystyle p_1,\dots ,p_n,}$ by the above property of the resultant.

This proves the weak Nullstellensatz by induction on the number of variables.

AGröbner basis is an algorithmic concept that was introduced in 1973 byBruno Buchberger. It is presently fundamental incomputational 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 contains1 if and only if itsreduced Gröbner basis (for anymonomial ordering) is1.
• The number of the common zeros of the polynomials in a Gröbner basis is strongly related to the number ofmonomials that areirreducibles 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 alexicographic monomial order, the common zeros can be computed by solving iterativelyunivariate polynomials (this is not used in practice since one knows better algorithms).
• Strong Nullstellensatz: a power ofp belongs to an idealI if and only thesaturation ofI byp produces the Gröbner basis1. Thus, the strong Nullstellensatz results almost immediately from the definition of the saturation.

The Nullstellensatz is subsumed by a systematic development of the theory ofJacobson 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 ifk is a field, then every finitely generatedk-algebraR (necessarily of the form$R=k[t_1,\cdots ,t_n]/I$) is Jacobson. More generally, one has the following theorem:

Let${\displaystyle R}$ be a Jacobson ring. If${\displaystyle S}$ is a finitely generatedR-algebra, then${\displaystyle S}$ is a Jacobson ring. Furthermore, if${\displaystyle \mathfrak{n}\subseteq S}$ is a maximal ideal, then${\displaystyle \mathfrak{m}:=\mathfrak{n}\cap R}$ is a maximal ideal of$R$, and${\displaystyle S/\mathfrak{n}}$ is a finite extension of${\displaystyle R/\mathfrak{m}}$.^[4]

Other generalizations proceed from viewing the Nullstellensatz inscheme-theoretic terms as saying that for any fieldk and nonzero finitely generatedk-algebraR, the morphism$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}R\to \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k$ admits asection étale-locally (equivalently, afterbase change along$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}L\to \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}k$ for some finite field extension$L/k$). In this vein, one has the following theorem:

Anyfaithfully flat morphism of schemes$f:Y\to X$locally of finite presentation admits aquasi-section, in the sense that there exists a faithfully flat and locallyquasi-finite morphism$g:X^{\prime }\to X$ locally of finite presentation such that the base change$f^{\prime }:Y{\times }_X{X}^{\prime }\to X^{\prime }$ of$f$ along$g$ admits a section.^[5] Moreover, if$X$ isquasi-compact (resp. quasi-compact andquasi-separated), then one may take$X^{\prime }$ to be affine (resp.$X^{\prime }$ affine and$g$ quasi-finite), and if$f$ issmooth surjective, then one may take$g$ to beétale.^[6]

Serge Lang gave an extension of the Nullstellensatz to the case of infinitely many generators:

Let$\kappa$ be aninfinite cardinal and let$K$ be an algebraically closed field whosetranscendence degree over itsprime subfield is strictly greater than${\displaystyle \kappa }$. Then for any set$S$ of cardinality$\kappa$, the polynomial ring$A=K[x_i{]}_{i\in S}$ satisfies the Nullstellensatz, i.e., for any ideal$J\subset A$ we have that${\displaystyle \sqrt{J}=\text{I}(\text{V}(J))}$.^[7]

In all of its variants, Hilbert's Nullstellensatz asserts that some polynomialg belongs or not to an ideal generated, say, byf₁, ...,f_k; we haveg =f^r in the strong version,g = 1 in the weak form. This means the existence or the non-existence of polynomialsg₁, ...,g_k such thatg =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 theg_i.

It is thus a rather natural question to ask if there is an effective way to compute theg_i (and the exponentr 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 theg_i: such a bound reduces the problem to a finitesystem of linear equations that may be solved by usuallinear algebra techniques. Any such upper bound is called aneffective Nullstellensatz.

A related problem is theideal 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 theg_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 theg_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.^[8] 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:^[9]

Letf₁, ...,f_s be polynomials inn ≥ 2 variables, of total degreed₁ ≥ ... ≥d_s. If there exist polynomialsg_i such thatf₁g₁ + ... +f_sg_s = 1, then they can be chosen such that

${\displaystyle \mathrm{deg}(f_i{g}_i)\le max(d_s,3)\prod _{j=1}^{min(n,s)-1}max(d_j,3).}$

This bound is optimal if all the degrees are greater than 2.

Ifd is the maximum of the degrees of thef_i, this bound may be simplified to

${\displaystyle max(3,d{)}^{min(n,s)}.}$

An improvement due to M. Sombra is^[10]

${\displaystyle \mathrm{deg}(f_i{g}_i)\le 2d_s\prod _{j=1}^{min(n,s)-1}{d}_j.}$

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

We can formulate a certain correspondence between homogeneous ideals of polynomials and algebraic subsets of a projective space, called theprojective Nullstellensatz, that is analogous to the affine one. To do that, we introduce some notations. Let${\displaystyle R=k[t_0,\dots ,t_n].}$ The homogeneous ideal,

${\displaystyle R_{+}=\underset{d⩾1}{⨁}{R}_d}$

is called themaximal homogeneous ideal (see alsoirrelevant ideal). As in the affine case, we let: for a subset${\displaystyle S\subseteq {\mathbb{P}}^n}$ and a homogeneous idealI ofR,

By${\displaystyle f=0\text{ on }S}$ we mean: for every homogeneous coordinates${\displaystyle (a_0:\cdots :a_n)}$ of a point ofS we have${\displaystyle f(a_0,\dots ,a_n)=0}$. This implies that the homogeneous components off are also zero onS and thus that${\displaystyle {\mathrm{I}}_{{\mathbb{P}}^n}(S)}$ is a homogeneous ideal. Equivalently,${\displaystyle {\mathrm{I}}_{{\mathbb{P}}^n}(S)}$ is the homogeneous ideal generated by homogeneous polynomialsf that vanish onS. Now, for any homogeneous ideal${\displaystyle I\subseteq R_{+}}$, by the usual Nullstellensatz, we have:

${\displaystyle \sqrt{I}={\mathrm{I}}_{{\mathbb{P}}^n}({\mathrm{V}}_{{\mathbb{P}}^n}(I)),}$

and so, like in the affine case, we have:^[11]

There exists an order-reversing one-to-one correspondence between proper homogeneous radical ideals ofR and subsets of${\displaystyle {\mathbb{P}}^n}$ of the form${\displaystyle {\mathrm{V}}_{{\mathbb{P}}^n}(I).}$ The correspondence is given by${\displaystyle {\mathrm{I}}_{{\mathbb{P}}^n}}$ and${\displaystyle {\mathrm{V}}_{{\mathbb{P}}^n}.}$

The Nullstellensatz also holds for the germs of holomorphic functions at a point of complexn-space${\displaystyle {\mathbb{C}}^n.}$ Precisely, for each open subset${\displaystyle U\subseteq {\mathbb{C}}^n,}$ let${\displaystyle {\mathcal{O}}_{{\mathbb{C}}^n}(U)}$ denote the ring of holomorphic functions onU; then${\displaystyle {\mathcal{O}}_{{\mathbb{C}}^n}}$ is asheaf on${\displaystyle {\mathbb{C}}^n.}$ The stalk${\displaystyle {\mathcal{O}}_{{\mathbb{C}}^n,0}}$ at, say, the origin can be shown to be aNoetherianlocal ring that is aunique factorization domain.

If${\displaystyle f\in {\mathcal{O}}_{{\mathbb{C}}^n,0}}$ is a germ represented by a holomorphic function${\displaystyle \tilde{f}:U\to \mathbb{C}}$, then let${\displaystyle V_0(f)}$ be the equivalence class of the set

${\displaystyle \left\{z\in U\mid \tilde{f}(z)=0\right\},}$

where two subsets${\displaystyle X,Y\subseteq {\mathbb{C}}^n}$ are considered equivalent if${\displaystyle X\cap U=Y\cap U}$ for some neighborhoodU of 0. Note${\displaystyle V_0(f)}$ is independent of a choice of the representative${\displaystyle \tilde{f}.}$ For each ideal${\displaystyle I\subseteq {\mathcal{O}}_{{\mathbb{C}}^n,0},}$ let${\displaystyle V_0(I)}$ denote${\displaystyle V_0(f_1)\cap \cdots \cap V_0(f_r)}$ for some generators${\displaystyle f_1,\dots ,f_r}$ ofI. It is well-defined; i.e., is independent of a choice of the generators.

For each subset${\displaystyle X\subseteq {\mathbb{C}}^n}$, let

${\displaystyle I_0(X)=\left\{f\in {\mathcal{O}}_{{\mathbb{C}}^n,0}\mid V_0(f)\supset X\right\}.}$

It is easy to see that${\displaystyle I_0(X)}$ is an ideal of${\displaystyle {\mathcal{O}}_{{\mathbb{C}}^n,0}}$ and that${\displaystyle I_0(X)=I_0(Y)}$ if${\displaystyle X\sim Y}$ in the sense discussed above.

Theanalytic Nullstellensatz then states:^[12] for each ideal${\displaystyle I\subseteq {\mathcal{O}}_{{\mathbb{C}}^n,0}}$,

${\displaystyle \sqrt{I}=I_0(V_0(I))}$

where the left-hand side is theradical ofI.

• 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

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• Polynomials
• Theorems in algebraic geometry

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