Hilbert's Nullstellensatz
Encyclopedia
Hilbert's Nullstellensatz (German
: "theorem of zeros," or more literally, "zero-locus-theorem" – see Satz
) is a theorem which establishes a fundamental relationship between geometry
and algebra
. This relationship is the basis of algebraic geometry
, an important branch of mathematics
. It relates algebraic set
s to ideals
in polynomial ring
s over algebraically closed field
s. This relationship was discovered by David Hilbert
who proved Nullstellensatz and several other important related theorems named after him (like Hilbert's basis theorem
).
s) and K be an algebraically closed field extension (such as the complex number
s), consider the polynomial ring
k[X1,X2,..., Xn] and let I be an ideal
in this ring. The affine variety V(I) defined by this ideal consists of all n-tuples x = (x1,...,xn) in Kn such that f(x) = 0 for all f in I. Hilbert's Nullstellensatz states that if p is some polynomial in k[X1,X2,..., Xn] which vanishes on the variety V(I), i.e. p(x) = 0 for all x in V(I), then there exists a natural number
r such that pr is in I.
An immediate corollary is the "weak Nullstellensatz": The ideal I in k[X1,X2,..., Xn] contains 1 if and only if the polynomials in I do not have any common zeros in Kn.
When k=K the "weak Nullstellensatz" may also be stated as follows:
if I is a proper ideal in K[X1,X2,..., Xn], then V(I) cannot be empty
, i.e. there exists a common zero for all the polynomials in the ideal. 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 that K be algebraically closed is essential here; the elements of the proper ideal (X2 + 1) in R[X] do not have a common zero.
With the notation common in algebraic geometry, the Nullstellensatz can also be formulated as
for every ideal J. Here,
denotes the radical
of J and I(U) is the ideal of all polynomials which vanish on the set U.
In this way, we obtain an order-reversing bijective correspondence between the affine varieties in Kn and the radical ideals of K[X1,X2,..., Xn]. 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.
Let
be a Jacobson ring
. If
is a finitely generated R-algebra
, then
is a Jacobson ring. Further, if 
is a maximal ideal, then 
is a maximal ideal of R, and 
is a finite extension field of 
.
Another generalization states that a faithfully flat morphism
locally of finite type with X quasi-compact has a quasi-section, i.e. there exists 
affine and faithfully flat and quasi-finite over X together with an X-morphism 
.
over 
the matrices satisfy various polynomials such as their minimal polynomial
s, which form a proper ideal (because they are not all zero, in which case the result is trivial); one might call this the characteristic ideal, by analogy with the characteristic polynomial
.
One then defines an eigenvector for a commutative algebra as a vector v such that for all
one has 
for a linear functional
This simply linearizes the definition of an eigenvalue, and is the correct definition for a common eigenvector, as if v is a common eigenvector, meaning
then the functional is defined as
(treating scalars as multiples of the identity matrix
, which has eigenvalue 1 for all vectors), and conversely an eigenvector for such a functional 
is a common eigenvector. Geometrically, the eigenvalue corresponds to the point in affine k-space with coordinates 
with respect to the basis given by 
Then the existence of an eigenvalue
is equivalent to the ideal generated by (the relations satisfied by) 
being non-empty, which exactly generalizes the usual proof of existence of an eigenvalue existing for a single matrix over an algebraically closed field by showing that the characteristic polynomial has a zero.
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....
: "theorem of zeros," or more literally, "zero-locus-theorem" – see Satz
Satz (disambiguation)
Satz is a German word and name, and may refer to:* Satz, a formal section in music analysis* "theorem" in German, conventionally used in the name of certain theorems. Note however that the word "Theorem" is also used in German, generally for more important results, and thus in a stricter...
) is a theorem which establishes a fundamental relationship between geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
. This relationship is the basis of algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, an important branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. It relates algebraic set
Algebraic set
In mathematics, an algebraic set over an algebraically closed field K is the set of solutions in Kn of a set of simultaneous equationsand so on up to...
s to ideals
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
in polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
s over algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
s. This relationship was discovered by David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
who proved Nullstellensatz and several other important related theorems named after him (like Hilbert's basis theorem
Hilbert's basis theorem
In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated. This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the...
).
Formulation
Let k be a field (such as the rational numberRational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s) and K be an algebraically closed field extension (such as the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s), consider the polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
k[X1,X2,..., Xn] and let I be an ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
in this ring. The affine variety V(I) defined by this ideal consists of all n-tuples x = (x1,...,xn) in Kn such that f(x) = 0 for all f in I. Hilbert's Nullstellensatz states that if p is some polynomial in k[X1,X2,..., Xn] which vanishes on the variety V(I), i.e. p(x) = 0 for all x in V(I), then there exists a natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
r such that pr is in I.
An immediate corollary is the "weak Nullstellensatz": The ideal I in k[X1,X2,..., Xn] contains 1 if and only if the polynomials in I do not have any common zeros in Kn.
When k=K the "weak Nullstellensatz" may also be stated as follows:
if I is a proper ideal in K[X1,X2,..., Xn], then V(I) cannot be empty
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
, i.e. there exists a common zero for all the polynomials in the ideal. This is the reason for the name of the theorem, which can be proved easily from the 'weak' form using the Rabinowitsch trick
Rabinowitsch trick
In mathematics, the Rabinowitsch trick, introduced by ,is a short way of proving the general case of the Hilbert Nullstellensatz from an easier special case, by introducing an extra variable....
. The assumption that K be algebraically closed is essential here; the elements of the proper ideal (X2 + 1) in R[X] do not have a common zero.
With the notation common in algebraic geometry, the Nullstellensatz can also be formulated as
for every ideal J. Here,
denotes the radicalRadical of an ideal
In commutative ring theory, a branch of mathematics, the radical of an ideal I is an ideal such that an element x is in the radical if some power of x is in I. A radical ideal is an ideal that is its own radical...
of J and I(U) is the ideal of all polynomials which vanish on the set U.
In this way, we obtain an order-reversing bijective correspondence between the affine varieties in Kn and the radical ideals of K[X1,X2,..., Xn]. In fact, more generally, one has a Galois connection
Galois connection
In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets . The same notion can also be defined on preordered sets or classes; this article presents the common case of posets. Galois connections generalize the correspondence...
between subsets of the space and subsets of the algebra, where "Zariski closure" and "radical of the ideal generated" are the closure operator
Closure operator
In mathematics, a closure operator on a set S is a function cl: P → P from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S....
s.
Generalization
This generalization is due to Bourbaki, and is the most general form of the Nullstellensatz.Let
be a Jacobson ringJacobson ring
In mathematics, in the realm of ring theory, a commutative ring with identity is said to be a Hilbert ring or a Jacobson ring if every prime ideal of the ring is an intersection of maximal ideals....
. If
is a finitely generated R-algebraAssociative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
, then
is a Jacobson ring. Further, if is a maximal ideal, then is a maximal ideal of R, and is a finite extension field of .Another generalization states that a faithfully flat morphism
locally of finite type with X quasi-compact has a quasi-section, i.e. there exists affine and faithfully flat and quasi-finite over X together with an X-morphism .Commuting matrices
The fact that commuting matrices have a common eigenvector – and hence by induction stabilize a common flag and are simultaneously triangularizable – can be interpreted as a result of the weak Nullstellensatz, as follows: commuting matrices form a commutative algebra over the matrices satisfy various polynomials such as their minimal polynomial
Minimal polynomial (linear algebra)
In linear algebra, the minimal polynomial of an n-by-n matrix A over a field F is the monic polynomial P over F of least degree such that P=0...
s, which form a proper ideal (because they are not all zero, in which case the result is trivial); one might call this the characteristic ideal, by analogy with the characteristic polynomial
Characteristic polynomial
In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....
.
One then defines an eigenvector for a commutative algebra as a vector v such that for all
one has for a linear functionalLinear functional
In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
This simply linearizes the definition of an eigenvalue, and is the correct definition for a common eigenvector, as if v is a common eigenvector, meaning
then the functional is defined as(treating scalars as multiples of the identity matrix
, which has eigenvalue 1 for all vectors), and conversely an eigenvector for such a functional is a common eigenvector. Geometrically, the eigenvalue corresponds to the point in affine k-space with coordinates with respect to the basis given by Then the existence of an eigenvalue
is equivalent to the ideal generated by (the relations satisfied by) being non-empty, which exactly generalizes the usual proof of existence of an eigenvalue existing for a single matrix over an algebraically closed field by showing that the characteristic polynomial has a zero.See also
- Stengle's PositivstellensatzStengle's PositivstellensatzIn real semialgebraic geometry, Stengle's Positivstellensatz characterizes polynomials that are positive on a semialgebraic set, which is defined by systems of inequalities of polynomials with real coefficients, or more generally, coefficients from any real-closed field.It can be thought of as an...
- Differential Nullstellensatz
- Combinatorial Nullstellensatz