Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Algebra and geometry

Hilbert’s Nullstellensatz

Start with an ideal of polynomials in finitely many variables and take its common zeros over an algebraically closed field. A polynomial vanishes on all those points exactly when some positive power of it lies in the original ideal.

Exact theorem

Exact Mathlib statement

theorem MvPolynomial.vanishingIdeal_zeroLocus_eq_radical {k K : Type*} [Field k] [Field K] [Algebra k K] {σ : Type*} [IsAlgClosed K] [Finite σ] (I : Ideal (MvPolynomial σ k)) : vanishingIdeal k (zeroLocus K I) = I.radical

The theorem at a glance

Hilbert’s Nullstellensatz at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

The ideal of all k-polynomials vanishing on the K-valued common zeros of I is exactly the radical of I. Explanatory diagram.
Detailed visual description

The poster centers the equality between the vanishing ideal of the zero locus and the radical. Below it, an algebra–geometry–algebra route passes through maximal ideals and realized points before closing both ideal inclusions.

Statement structure

Statement and scope

Statement map for Hilbert’s NullstellensatzFor an ideal of polynomials in finitely many variables over k, evaluated at K-valued points with K algebraically closed, the polynomials vanishing on all common zeros form exactly the radical ideal. Claim boundary: This target indexes Mathlib's strong affine Nullstellensatz for finitely many variables: coefficients lie in a field k, zeros are evaluated in an algebraically closed k-algebra field K, and the returned k-polynomial ideal is the radical of I. It supplies no effective exponent, algorithm, multiplicity statement, or infinite-variable, projective, or scheme-level theorem. The pinned upstream declaration is MvPolynomial.vanishingIdeal_zeroLocus_eq_radical. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingFor an ideal ofpolynomials in finitelymany variables over k,evaluated at K-valuedpoints with Kalgebraically closed,the polynomialsvanishing on all commonzeros form exactly theradical ideal.Claim boundaryThis target indexesMathlib's strong affineNullstellensatz forfinitely many variables:coefficients lie in afield k, zeros areevaluated in analgebraically closedk-algebra field K, andthe returnedk-polynomial ideal isthe radical of I. Itsupplies no effectiveexponent, algorithm,multiplicity statement,or infinite-variable,projective, orscheme-level theorem.Pinned declarationmathlib ·MvPolynomial.vanishingIdeal_zeroLocus_eq_radicalStatement map for Hilbert’s NullstellensatzFor an ideal of polynomials in finitely many variables over k, evaluated at K-valued points with K algebraically closed, the polynomials vanishing on all common zeros form exactly the radical ideal. Claim boundary: This target indexes Mathlib's strong affine Nullstellensatz for finitely many variables: coefficients lie in a field k, zeros are evaluated in an algebraically closed k-algebra field K, and the returned k-polynomial ideal is the radical of I. It supplies no effective exponent, algorithm, multiplicity statement, or infinite-variable, projective, or scheme-level theorem. The pinned upstream declaration is MvPolynomial.vanishingIdeal_zeroLocus_eq_radical. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingFor an ideal ofpolynomials in finitelymany variables over k,evaluated at K-valuedpoints with Kalgebraically closed,the polynomialsvanishing on all commonzeros form exactly theradical ideal.Claim boundaryThis target indexesMathlib's strong affineNullstellensatz forfinitely many variables:coefficients lie in afield k, zeros areevaluated in analgebraically closedk-algebra field K, andthe returnedk-polynomial ideal isthe radical of I. Itsupplies no effectiveexponent, algorithm,multiplicity statement,or infinite-variable,projective, orscheme-level theorem.Pinned declarationmathlib ·MvPolynomial.vanishingIdeal_zeroLocus_eq_radical

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

Passing from equations to common zeros and back recovers exactly the radical ideal. Explanatory diagram.
Detailed visual description

An order-reversing algebra-to-geometry-to-algebra loop begins with a nested ideal, opens into a finite-coordinate zero locus, and closes at a softened radical shell around the starting ideal. Multiple maximal-ideal branches meet individual points without suggesting a bijection.

Why it matters

A mathematical landmark

Hilbert's Nullstellensatz is the central algebra–geometry bridge of affine algebraic geometry. Mathlib's declaration makes the two coefficient fields and the finite-variable hypothesis explicit while identifying the algebraic closure operation exactly.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally
Locally reproducedExact upstream declaration replayed
Reviewed pageCurrent public presentation reviewed
Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This target indexes Mathlib's strong affine Nullstellensatz for finitely many variables: coefficients lie in a field k, zeros are evaluated in an algebraically closed k-algebra field K, and the returned k-polynomial ideal is the radical of I. It supplies no effective exponent, algorithm, multiplicity statement, or infinite-variable, projective, or scheme-level theorem.

Source and local evidence

Where the theorem comes from

Existing declaration
MvPolynomial.vanishingIdeal_zeroLocus_eq_radical in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.hilbert-nullstellensatz.v001
Source
Open the pinned upstream reference