Hilbert's Nullstellensatz

related topics
{math, number, function}

Hilbert's Nullstellensatz (German: "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem which makes precise a fundamental relationship between the geometric and algebraic sides of algebraic geometry, an important branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. The theorem was first proved by David Hilbert, after whom it is named.

Contents

Formulation

Let k be a field (such as the rational numbers) and K be an algebraically closed field extension (such as the complex numbers), 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. The weak Nullstellensatz is a generalization of the fundamental theorem of algebra, which is the case n = 1, I in this case being the proper ideal generated by the polynomial, and V(I) being the zeros.

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

for every ideal J. Here, \sqrt{J} denotes the radical of J and I(U) is the ideal of all polynomials which vanish on the set U.

Full article ▸

related documents
Linearity of integration
Rectangle
Sigmoid function
Constant folding
Z notation
List of Fourier-related transforms
Euler's theorem
Derivative of a constant
Direct sum of groups
Essential singularity
Discrete mathematics
Group object
Product of group subsets
Greibach normal form
Surjective function
Online algorithm
The Third Manifesto
Distinct
De Bruijn-Newman constant
Hurwitz polynomial
Context-free language
Precondition
CycL
Short five lemma
Conjugate closure
Dense set
RC5
Recursive language
Lazy initialization
Data element