{
  "id": "rp.homogeneous-vanishing-fn",
  "wikidata_q": null,
  "item_type": "law",
  "name": {
    "ja": "有限体F上の斉次多項式で、次数dが|F|以下かつF^nの全ての点で0になるものは、ゼロ多項式である",
    "en": null
  },
  "summary": {
    "ja": "P_h ∈ MvPolynomial (Fin n) Fが次数dの斉次多項式でd ≤ |F|かつ全てのv ∈ F^nでeval v P_h = 0ならば、P_h = 0。Dvir 2009 JAMS版のTheorem 1.5の証明で使われる、斉次多項式が消えることの議論にあたる（arXiv版v3では対応するのはTheorem 3で、下界の式はC(q+n−2, n)、JAMS版のTheorem 1.5のC(q+n−1, n)と番号・式ともに異なる。Dvirの論文には「sub-lemma 5」という番号付けは無い。arXiv v3の番号付き命題はTheorem 1, 2, 3, Corollary 1.1, Lemma 2.1, Claim 2.2）。「sub-lemma 5」はRei-AIOS側のLean file `data/lean4-mathlib/CollatzRei/KakeyaDvir.lean`内部でPhase 2aの5件の分解項目の#5として付けた命名。Lean 4でmathlibの `MvPolynomial.IsHomogeneous.eq_zero_of_forall_eval_eq_zero_of_le_card` を直接適用して閉じている。依存する公理はpropext, Classical.choice, Quot.sound（mathlib base）。",
    "en": null
  },
  "formula": "∀ (F : Type*) [Field F] [Fintype F] [DecidableEq F] (n : ℕ) (P_h : MvPolynomial (Fin n) F) (d : ℕ), P_h.IsHomogeneous d → d ≤ Fintype.card F → (∀ v : Fin n → F, MvPolynomial.eval v P_h = 0) → P_h = 0",
  "expression_latex": "P_h \\in F[x_1, \\ldots, x_n] \\text{ homogeneous of degree } d \\;\\land\\; d \\le |F| \\;\\land\\; \\forall v \\in F^n.\\, P_h(v) = 0 \\;\\Longrightarrow\\; P_h = 0",
  "symbols": [
    {
      "name": "F",
      "meaning": "有限体",
      "type": "Type (Field + Fintype + DecidableEq)"
    },
    {
      "name": "n",
      "meaning": "変数の個数 = 空間F^nの次元",
      "type": "ℕ"
    },
    {
      "name": "MvPolynomial (Fin n) F",
      "meaning": "F係数n変数多項式環（mathlib）",
      "type": "Type"
    },
    {
      "name": "P_h",
      "meaning": "斉次多項式（homogeneous polynomial）",
      "type": "MvPolynomial (Fin n) F"
    },
    {
      "name": "d",
      "meaning": "斉次次数（homogeneous degree）",
      "type": "ℕ"
    },
    {
      "name": "IsHomogeneous d",
      "meaning": "多項式が全次数dで斉次（mathlibの述語）",
      "type": "MvPolynomial (Fin n) F → ℕ → Prop"
    },
    {
      "name": "Fintype.card F",
      "meaning": "有限集合Fの要素数 = |F|",
      "type": "ℕ"
    },
    {
      "name": "MvPolynomial.eval v",
      "meaning": "多項式を点v ∈ F^nで評価（mathlib）",
      "type": "(Fin n → F) → MvPolynomial (Fin n) F → F"
    }
  ],
  "example": {
    "description": {
      "ja": "Lean 4で `homogeneous_vanishing_on_Fn_is_zero` を証明し、`#print axioms` でaxiom profileを出力する。本定理はRei-AIOS側のKakeyaDvir.lean内部で付けた5件のsub-lemma分解項目の#5（Dvir 2009 JAMS版のTheorem 1.5証明で使われる斉次多項式消失の議論にあたる、arXiv版v3では対応するのはTheorem 3の証明中の斉次多項式消失の議論、Dvirの論文にはsub-lemma番号付けなし）。同じKakeyaDvir.leanにRei-AIOS側のsub-lemma 1-4（= sorry残の未完成項目）とmain theorem `dvir_kakeya`（= sub-lemma 1-4に依存するためsorryAx残）も含まれる。本項目はmathlib baseのみに依存し、ファイル内の他のsorryには影響されない。",
      "en": null
    },
    "code": "import CollatzRei.KakeyaDvir\nopen CollatzRei.KakeyaDvir\n#check @homogeneous_vanishing_on_Fn_is_zero\n#print axioms homogeneous_vanishing_on_Fn_is_zero",
    "expected_output": "@homogeneous_vanishing_on_Fn_is_zero : ∀ {F : Type u_1} [inst : Field F] [inst_1 : Fintype F] [DecidableEq F] {n : ℕ}\n  (P_h : MvPolynomial (Fin n) F) (d : ℕ),\n  P_h.IsHomogeneous d → d ≤ Fintype.card F → (∀ (v : Fin n → F), (MvPolynomial.eval v) P_h = 0) → P_h = 0\n'CollatzRei.KakeyaDvir.homogeneous_vanishing_on_Fn_is_zero' depends on axioms: [propext, Classical.choice, Quot.sound]"
  },
  "sources": [
    {
      "title": "On the size of Kakeya sets in finite fields",
      "author": "Dvir, Z.",
      "year": 2009,
      "url": "https://doi.org/10.1090/S0894-0347-08-00607-3",
      "wayback_url": null,
      "retrieved_at": null,
      "license": null,
      "license_note": "未確認（出版社AMS、Journal of the AMS 22(4), 2009, 1093–1097; プレプリント2008）",
      "note": "下界C(q+n−1, n)の根拠としてTheorem 1.5の証明を参照（AlonとTaoの指摘による改良と記す）。独立レビュアーがJAMS版のPDFで確認（2026-10-16）：Theorem 1.5（1.1節）、証明は3節で|K| < C(q+n−1, n)を仮定"
    },
    {
      "title": "On the size of Kakeya sets in finite fields (arXiv preprint, open-access)",
      "author": "Dvir, Z.",
      "year": 2008,
      "url": "https://arxiv.org/abs/0803.2336",
      "wayback_url": null,
      "retrieved_at": null,
      "license": null,
      "license_note": "未確認（arXiv）",
      "note": "この版では対応する結果はTheorem 3で、証明はC(q+n−2, n)を使う。定理の番号と式はJAMS版と異なる。Theorem 3の証明中、斉次成分P_{q−1}が全てのyで消えるので恒等的に零と結論する段落はarXiv v3、1.1節のTheorem 3の証明に含まれる（起草sessionがarXiv v3 PDFで直接verify、2026-10-16）"
    },
    {
      "title": "Mathlib.RingTheory.MvPolynomial.Homogeneous (mathlib4) — `IsHomogeneous.eq_zero_of_forall_eval_eq_zero_of_le_card`",
      "author": "mathlib4 contributors",
      "year": 2024,
      "url": "https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/RingTheory/MvPolynomial/Homogeneous.lean",
      "wayback_url": null,
      "retrieved_at": null,
      "license": "Apache-2.0 (mathlib4)"
    }
  ],
  "relations": [
    {
      "kind": "related_to",
      "target_id": "rp.dvir-bound-3-2",
      "note": "本定理はRei-AIOS側でDvirの証明を分解した5項目の#5（Dvir 2009（JAMS版）証明の構成要素）。同ファイルKakeyaDvir.leanに下界の値の計算項目（dvirBound_3_2等）もあり、両者は同じ証明鎖の別要素"
    },
    {
      "kind": "related_to",
      "target_id": "rp.dvir-bound-3-4",
      "note": "両者ともDvir 2009（JAMS版）証明の構成要素（本項目は斉次多項式消失で、Rei-AIOS側でDvirの証明を分解した5項目の#5。target_idは下界の値の計算）"
    },
    {
      "kind": "related_to",
      "target_id": "rp.dvir-bound-5-4",
      "note": "両者ともDvir 2009（JAMS版）証明の構成要素（本項目は斉次多項式消失で、Rei-AIOS側でDvirの証明を分解した5項目の#5。target_idは下界の値の計算）"
    }
  ],
  "verify_source_theorem": "CollatzRei.KakeyaDvir.homogeneous_vanishing_on_Fn_is_zero",
  "verify_source_file": "CollatzRei/KakeyaDvirAxiomCheck.lean",
  "authors": [
    {
      "kind": "bot",
      "name": "reipedia-renderer",
      "model": "deterministic-template-step2343",
      "contribution": "Generated HTML + JSON from source item JSON via Astro static build",
      "at": "2026-10-15T00:00:00+09:00"
    },
    {
      "kind": "AI",
      "name": "Claude （Rei-AIOS）",
      "model": "claude-opus-4-7",
      "contribution": "項目本文（一行説明・式・記号表・走る最小例・出典・関係リンク）を起草",
      "at": "2026-10-15T00:00:00+09:00"
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    {
      "kind": "AI",
      "name": "Claude （Rei-AIOS）",
      "model": "claude-opus-4-7",
      "contribution": "下書きの改訂（表記規則の適用、英語用語の日本語化、公開前文言の削除、関係項目の追加、走る最小例の期待出力をLeanの実行結果に合わせる、license_noteの表記修正）",
      "at": "2026-10-28T00:00:00+09:00"
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    {
      "kind": "human",
      "name": "藤本 伸樹",
      "contribution": "5項目の下書き範囲を指定",
      "at": "2026-10-15T00:00:00+09:00"
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  ],
  "authorship": {
    "created_by": [
      {
        "agent": "rei-aios-session",
        "model": "claude-opus-4-7",
        "session": "rei-aios-77",
        "role": "draft",
        "date": "2026-10-15"
      },
      {
        "agent": "rei-aios-session",
        "model": "claude-opus-4-7",
        "session": "rei-aios-1b",
        "role": "revise",
        "date": "2026-10-28"
      }
    ],
    "reviewed_by": [
      {
        "agent": "claude-code-subagent",
        "kind": "independent",
        "session": "claude-code-subagent:rei-aios-1b:homogeneous-vanishing-fn-review:2026-10-28",
        "date": "2026-10-28"
      }
    ],
    "human_approver": "fujimoto-nobuki",
    "approval_date": "2026-10-28"
  },
  "inspected_at": null,
  "reinspect_due": null,
  "inspector_error_rate": null,
  "inspection": {
    "state": "S2",
    "state_label": "形式証明済み (mathlib base)",
    "axioms": [
      "Classical.choice",
      "Quot.sound",
      "propext"
    ],
    "is_zero_axiom": false,
    "is_mathlib_base": true,
    "has_sorry_ax": false,
    "has_native_decide": false,
    "has_user_axiom": false,
    "source": "gha",
    "generated_at": "2026-10-18T23:15:15.522Z",
    "gha_run_id": "39175017999",
    "lean_toolchain": "leanprover/lean4:v4.27.0",
    "verify_source_theorem": "CollatzRei.KakeyaDvir.homogeneous_vanishing_on_Fn_is_zero",
    "verify_source_file": "CollatzRei/KakeyaDvirAxiomCheck.lean"
  }
}