{ "artifactId": "artifact.library.mathlib.hilbert-nullstellensatz.v001", "candidateOnly": false, "category": "Algebra and geometry", "claimBoundary": "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.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "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", "family": "Affine algebraic geometry", "id": "library.mathlib.hilbert-nullstellensatz.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/hilbert-nullstellensatz/", "entryData": "/data/library-theorems/hilbert-nullstellensatz.json", "evidence": "/proofs/artifact.library.mathlib.hilbert-nullstellensatz.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.hilbert-nullstellensatz.v001.evidence.json", "source": "/sources/upstream/hilbert-nullstellensatz/" }, "landmark": { "completeFormalType": "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", "concepts": [ "multivariable polynomials", "ideals", "zero loci", "vanishing ideals", "radicals", "algebraically closed fields" ], "plainLanguage": "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.", "poster": { "alt": "An ivory algebraic-geometry poster traces an ideal to its zero locus and back to its radical, with finite-variable and algebraically closed field hypotheses.", "byteSize": 3108959, "caption": "The ideal of all k-polynomials vanishing on the K-valued common zeros of I is exactly the radical of I.", "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.", "derivatives": [ { "byteSize": 146074, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/hilbert-nullstellensatz/theorem-poster-v1-640.webp", "sha256": "sha256:8bb7045f50e722009479cec4fb34e3af921e3732e7a1bd0850f946fe53dbc897", "width": 640 }, { "byteSize": 331534, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/hilbert-nullstellensatz/theorem-poster-v1-1024.webp", "sha256": "sha256:98a784982f810c142bf08889477b557035ed1ed07a416179dc1eb870a6644230", "width": 1024 } ], "generatedAt": "2026-07-25T06:04:24.711Z", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/hilbert-nullstellensatz/theorem-poster-v1.png", "sha256": "sha256:eb77d0952b04699c911947a82704a9cce48e6355abde9445979bb526b1e78263", "title": "Hilbert’s Nullstellensatz at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "ALGEBRAIC GEOMETRY · EQUATIONS AND ZERO SETS\nHILBERT’S NULLSTELLENSATZ\nk and K fields · K algebraically closed · finitely many variables\nIₖ(V_K(I)) = √I\nEQUATIONS → COMMON ZEROS → RADICAL IDEAL\nHOW THE CHECKED ROUTE MOVES\n1 · Show the radical vanishes on every common zero\n2 · Rewrite the radical as an intersection of maximal ideals\n3 · Realize each maximal ideal by a K-valued point\n4 · Use that point to prove the reverse containment\nEXACT SCOPE\nAffine finite-variable theorem. No effective bound, algorithm, or point–ideal bijection." }, "visual": { "alt": "An ideal of k-polynomials in finitely many variables maps to its K-valued common zero locus, where K is algebraically closed, and returns through the vanishing-ideal operation as the radical of the original ideal.", "byteSize": 3125457, "caption": "Passing from equations to common zeros and back recovers exactly the radical ideal.", "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.", "derivatives": [ { "byteSize": 55438, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/hilbert-nullstellensatz/theorem-schematic-v1-640.webp", "sha256": "sha256:f692c7c9d13ff538418e22ce944f7b8196271ec1ac63a6fad21fdbae41417610", "width": 640 }, { "byteSize": 181958, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/hilbert-nullstellensatz/theorem-schematic-v1-1200.webp", "sha256": "sha256:618f1b0fb43000d1d8453218255d476803df76e4445a6ff747ed0513e313c693", "width": 1200 } ], "generatedAt": "2026-07-25T05:27:59.621Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/hilbert-nullstellensatz/theorem-schematic-v1.png", "sha256": "sha256:0c0352de762f3df3562d8d540770be21f9b4d7ff7f97df89d1a4df81c911d0bf", "title": "Hilbert’s Nullstellensatz schematic", "width": 1536 }, "walkthrough": null, "whyLandmark": "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." }, "nonClaims": [ "Proof Atlas did not originate Hilbert's Nullstellensatz or Mathlib's declaration.", "The theorem does not say I itself equals its vanishing ideal unless I is radical, and it does not assert a point–maximal-ideal bijection in the stated two-field generality.", "The generated explanation and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.hilbert-nullstellensatz.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "Current public presentation reviewed", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "For 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.", "targetId": "target.library.mathlib.hilbert-nullstellensatz.v001", "title": "Hilbert’s Nullstellensatz", "upstreamOrigin": { "declarationName": "MvPolynomial.vanishingIdeal_zeroLocus_eq_radical", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/RingTheory/Nullstellensatz.lean#L171", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:cc19eaf5462c113cef15fdce99f41799b616333cdc33cc0ffaac2086e1bdf2c4", "sourceByteLength": 7929, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/RingTheory/Nullstellensatz.lean", "sourceLine": 171, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }