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 sets to ideals in polynomial rings over algebraically closed fields. 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).
Let
k
K
k
k[X1,\ldots,Xn]
I
V(I)
n
x=(x1,...,xn)
Kn
f(x)=0
f
k[X1,\ldots,Xn]
V(I)
p(x)=0
x
V(I)
r
pr
I
An immediate corollary is the weak Nullstellensatz: The ideal
I\subseteqk[X1,\ldots,Xn]
k[X1,\ldots,Xn],
\R[X]
\R.
With the notation common in algebraic geometry, the Nullstellensatz can also be formulated as
\hbox{I}(\hbox{V}(J))=\sqrt{J}
for every ideal J. Here,
\sqrt{J}
In this way, taking
k=K
K[X1,\ldots,Xn].
As a particular example, consider a point
P=(a1,...,an)\inKn
I(P)=(X1-a1,\ldots,Xn-an)
\sqrt{I}=
cap | |
(a1,...,an)\inV(I) |
(X1-a1,...,Xn-an).
Conversely, every maximal ideal of the polynomial ring
K[X1,\ldots,Xn]
K
(X1-a1,\ldots,Xn-an)
a1,\ldots,an\inK
As another example, an algebraic subset W in Kn is irreducible (in the Zariski topology) if and only if
I(W)
There are many known proofs of the theorem. Some are non-constructive, such as the first one. Others are constructive, as based on algorithms for expressing or as a linear combination of the generators of the ideal.
Zariski's lemma asserts that if a field is finitely generated as an associative algebra over a field, then it is a finite field extension of (that is, it is also finitely generated as a vector space).
Here is a sketch of a proof using this lemma.[2]
Let
A=k[t1,\ldots,tn]
kn
\sqrt{I}\subseteqI(V)
f\not\in\sqrt{I}
f\not\inak{p}
ak{p}\supseteqI
R=(A/ak{p})[f-1]
ak{m}
R
R/ak{m}
xi
ti
A\tok
R
x=(x1,\ldots,xn)\inV
f(x)\ne0
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 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
p1,\ldots,pn,
p1
n-1
u2,\ldots,un,
R=\operatorname{Res}x(p1,u2p2+ … +unpn).
As is in the ideal generated by
p1,\ldots,pn,
u2,\ldots,un.
p1,\ldots,pn.
p1,\ldots,pn,
This proves the weak Nullstellensatz by induction on the number of variables.
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:
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 ) is Jacobson. More generally, one has the following theorem:
Let
R
S
S
ak{n}\subseteqS
ak{m}:=ak{n}\capR
S/ak{n}
R/ak{m}
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 admits a section étale-locally (equivalently, after base change along for some finite field extension ). In this vein, one has the following theorem:
Any faithfully flat morphism of schemes locally of finite presentation admits a quasi-section, in the sense that there exists a faithfully flat and locally quasi-finite morphism locally of finite presentation such that the base change of along admits a section.[4] Moreover, if is quasi-compact (resp. quasi-compact and quasi-separated), then one may take to be affine (resp. affine and quasi-finite), and if is smooth surjective, then one may take to be étale.[5] Serge Lang gave an extension of the Nullstellensatz to the case of infinitely many generators:
Let be an infinite cardinal and let be an algebraically closed field whose transcendence degree over its prime subfield is strictly greater than
\kappa
\sqrt{J}=\hbox{I}(\hbox{V}(J))
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:
Let be polynomials in variables, of total degree . If there exist polynomials such that, then they can be chosen such that
\deg(figi)\lemax(ds,3)\prod
min(n,s)-1 | |
j=1 |
max(dj,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)min(n,s).
An improvement due to M. Sombra is
\deg(figi)\le2ds\prod
min(n,s)-1 | |
j=1 |
dj.
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 the projective Nullstellensatz, that is analogous to the affine one. To do that, we introduce some notations. Let
R=k[t0,\ldots,tn].
R+=oplusdRd
is called the maximal homogeneous ideal (see also irrelevant ideal). As in the affine case, we let: for a subset
S\subseteqPn
\begin{align} \operatorname{I} | |
Pn |
(S)&=\{f\inR+\midf=0onS\},
\\ \operatorname{V} | |
Pn |
(I)&=\{x\inPn\midf(x)=0forallf\inI\}. \end{align}
By
f=0onS
(a0: … :an)
f(a0,\ldots,an)=0
\operatorname{I} | |
Pn |
(S)
\operatorname{I} | |
Pn |
(S)
I\subseteqR+
\sqrt{I}=
\operatorname{I} | |
Pn |
(\operatorname{V} | |
Pn |
(I)),
and so, like in the affine case, we have:[7]
There exists an order-reversing one-to-one correspondence between proper homogeneous radical ideals of R and subsets of
Pn
\operatorname{V} | |
Pn |
(I).
\operatorname{I} | |
Pn |
\operatorname{V} | |
Pn |
.
The Nullstellensatz also holds for the germs of holomorphic functions at a point of complex n-space
\Complexn.
U\subseteq\Complexn,
l{O} | |
\Complexn |
(U)
l{O} | |
\Complexn |
\Complexn.
l{O} | |
\Complexn,0 |
If
f\in
l{O} | |
\Complexn,0 |
\widetilde{f}:U\to\Complex
V0(f)
\left\{z\inU\mid\widetilde{f}(z)=0\right\},
where two subsets
X,Y\subseteq\Complexn
X\capU=Y\capU
V0(f)
\widetilde{f}.
I\subseteq
l{O} | |
\Complexn,0 |
,
V0(I)
V0(f1)\cap...\capV0(fr)
f1,\ldots,fr
For each subset
X\subseteq\Complexn
I0(X)=\left\{f\in
l{O} | |
\Complexn,0 |
\midV0(f)\supsetX\right\}.
It is easy to see that
I0(X)
l{O} | |
\Complexn,0 |
I0(X)=I0(Y)
X\simY
The analytic Nullstellensatz then states: for each ideal
I\subseteq
l{O} | |
\Complexn,0 |
\sqrt{I}=I0(V0(I))
where the left-hand side is the radical of I.
. Mukai . Shigeru Mukai . William Oxbury (trans.) . An Introduction to Invariants and Moduli . Cambridge studies in advanced mathematics . 81 . 2003 . 0-521-80906-1 . 82.