From polynomials to geometry — zero sets in affine space and their coordinate rings
An affine variety is the set of common zeros of a collection of polynomials in affine space kn. Given an ideal I ⊆ k[x1, …, xn], the variety V(I) consists of all points where every polynomial in I vanishes. This simple definition — "where do the polynomials equal zero?" — is the foundation of algebraic geometry.
The coordinate ring k[V] = k[x1, …, xn]/I(V) captures the algebraic structure of a variety. It is the ring of polynomial functions on V, and its algebraic properties (prime ideals, dimension, nilpotents) reflect the geometric properties of V. This tight correspondence between algebra and geometry — explored further in the Rings & Fields module — is what makes algebraic geometry so powerful.
Select a curve preset to visualize V(f) — the zero set of a polynomial f(x, y) in the affine plane. The contour is rendered using marching squares.
The zero locus V(f) is rendered using marching squares on a fine grid. Select a preset to explore different plane algebraic curves.
For a single polynomial f ∈ k[x, y], the variety V(f) is a curve in the plane — a parabola, an ellipse, or something more exotic. For an ideal I = (f1, …, fr), the variety V(I) is the intersection of the individual zero sets. The map V sends larger ideals to smaller varieties: if I ⊆ J, then V(J) ⊆ V(I). This order-reversing correspondence is the first hint of the deep duality between algebra and geometry.
The Zariski topology on kn declares the varieties V(I) to be the closed sets. Unlike the Euclidean topology, Zariski-open sets are large and dense — the complement of a hypersurface is open, and nonempty open sets are never disjoint. This coarse topology is perfectly adapted to algebraic (rather than analytic) questions.
The coordinate ring k[V] is the quotient k[x1, …, xn]/I(V), where I(V) is the ideal of all polynomials vanishing on V. Two polynomials define the same function on V precisely when their difference lies in I(V). The ring k[V] is a finitely generated reduced k-algebra, and conversely every such algebra arises as the coordinate ring of some affine variety.
Geometric properties translate directly: V is irreducible if and only if k[V] is an integral domain. The dimension of V equals the Krull dimension of k[V]. Points of V correspond to maximal ideals of k[V] (when k is algebraically closed). This dictionary — the foundation of the Nullstellensatz lesson — turns geometric intuition into algebraic computation and vice versa.