nLab zero locus

Idea

The zero locus or vanishing locus of a function is the set of points where it vanishes, in that it takes the value zero.

Definition

For $f \colon X \to \mathbb{A}$ a function, its zero locus is the preimage $f^{-1}(0)$ of zero, hence the level set at 0.

Properties

Nullstellensatz

Hilbert’s Nullstellensatz (German: “zero locus theorem”) characterizes joint zero loci of ideals of functions in a polynomial ring.

Related concepts

• quadratic formula

• critical locus, derived critical locus

• moment map, Hamiltonian

• zero-set structure