Nullsatz

Nullsatz, in many German-language texts abbreviated for Hilbert's Nullstellensatz, is a central theorem of algebraic geometry and commutative algebra that connects algebraic and geometric objects. The theorem is formulated for an algebraically closed field k and a polynomial ring k[x1, ..., xn]. For an ideal I ⊆ k[x1, ..., xn], let V(I) denote the set of common zeros in k^n.

The Nullstellensatz has several forms, including:

- Weak Nullstellensatz: If I is a proper ideal of k[x1, ..., xn], then V(I) ≠ ∅. Equivalently, if V(I) = ∅, then I = k[x1, ..., xn].

- Strong Nullstellensatz: I(V(I)) = sqrt(I), where I(V(I)) = {f ∈ k[x1, ..., xn] | f(a) = 0 for all a ∈ V(I)} and sqrt(I) is the radical of I (the set of polynomials whose some power lies in I).

Consequences: There is a natural one-to-one correspondence between radical ideals of k[x1, ..., xn] and affine algebraic sets in k^n. Morphisms of varieties correspond to quotient maps between coordinate rings, yielding a duality between geometric and algebraic descriptions.

Variants and extensions include the Real Nullstellensatz for real closed fields and various effective forms used in symbolic computation, such as those based on Gröbner bases.

Historically, Hilbert established the Nullstellensatz in the late 19th century as a foundational result in algebraic geometry. In German texts it is commonly referred to as der Nullstellensatz or simply der Nullsatz.

This article provides a concise overview; for detailed proofs and applications, see standard expositions in algebraic geometry and commutative algebra.