{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-polynomial",
  "semantic_family_id": "sf-polynomial",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "一元有理係數多項式"
  },
  "summary": {
    "zh-Hant": "把多項式視為有限支撐的有理係數資料；合併同次項後再讀取係數與次數。"
  },
  "statement": {
    "zh-Hant": "Q[x] 的一元多項式以有限個有理係數 a_k 表示 p(x)=a_0+a_1x+⋯+a_nx^n，指數均為非負整數。非零多項式的次數為最大非零係數的索引；本條目把零多項式的次數記為未定義。"
  },
  "formula": {
    "tex": "x^{n}",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mi>x</mi><mi>n</mi></msup></math>",
    "semantic_ast": {
      "type": "power",
      "base": {
        "type": "symbol",
        "name": "x",
        "glyph": "x"
      },
      "exponent": {
        "type": "symbol",
        "name": "n",
        "glyph": "n"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "f170087fad13cf934e7a5794678424d7d7bc52f3150c291125416b1439ac5baf"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "固定形式變數 x，係數屬於 Q，僅有限個係數非零。"
    },
    {
      "id": "a2",
      "zh-Hant": "係數向量按 [a_0,a_1,…] 由低次至高次排列；尾端零不改變多項式。"
    }
  ],
  "symbols": [
    {
      "token": "x",
      "role": "indeterminate",
      "role_zh": "形式變數",
      "scope": "representative_formula"
    },
    {
      "token": "n",
      "role": "natural_exponent",
      "role_zh": "代表單項式的非負整數指數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-variable-expression",
      "title_zh": "變數與代數式",
      "reason_zh": "先區分形式表達式與指定代入值。"
    },
    {
      "id": "mko-power-root",
      "title_zh": "自然數冪與主平方根",
      "reason_zh": "多項式項使用非負整數次冪。"
    },
    {
      "id": "mko-rational-number",
      "title_zh": "有理數",
      "reason_zh": "係數明示在有理數域中。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "多項式不是只看字面項數。相同次冪的係數要先相加；例如 x²−x²+3 化成常數 3，次數為零。零多項式與非零常數多項式不可混淆。",
      "加法逐次相加係數；乘法按指數和合併，所得係數是有限卷積。代入一個數值是對同一份係數資料求值，單個代入值相等不建立多項式恆等式。",
      "本輪題庫限定整數係數與整數輸入的精確切片，係數查詢超過支撐處為零。零多項式的『未定義』是教學慣例；未來須分別對應 Mathlib.degree 的 ⊥ 與 natDegree 的補零，不能混作一般次數。"
    ],
    "common_misconception_zh": "未先合併同次項就取最高字面指數，或用向量長度及尾端零決定次數。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限係數可逐項驗證；本條目尚未形式化多項式資料、求值與次數的語義橋。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "綁定 Polynomial Q、coeff、eval、degree 與 natDegree 的零多項式差異。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-polynomial.md 整理；2026-10-03 脈衝 R02，英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-03"
  }
}