{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R155",
  "content_hash": null,
  "slug": "cmi-claim-fiber-profile",
  "type": "claim",
  "title": "The complete fiber profile follows from one character sum",
  "summary": "Exactly (p-1)/2 values have one preimage, 1+chi values have two, and (p-3-4chi)/6 have three.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "For \\(r=0,1,2,3\\), let \\(N_r\\) count the values with exactly \\(r\\) preimages under \\(f(x)=x^3+x\\), and put \\(\\chi=\\left(\\frac{-3}{p}\\right)\\).\n\nA fiber has two distinct points precisely when \\(f(x)-y\\) has a repeated root. The derivative is \\(3x^2+1\\), which has \\(1+\\chi\\) roots. If \\(a\\) is one of them, then \\(f(a)=2a/3\\) and\n\\[\nf(x)-f(a)=(x-a)^2(x+2a).\n\\]\nThe two critical values are distinct when they exist, so \\(N_2=1+\\chi\\).\n\nFor distinct \\(x,z\\), the equality \\(f(x)=f(z)\\) is equivalent to\n\\[\nx^2+xz+z^2+1=0.\n\\]\nFor fixed \\(z\\), this quadratic has \\(1+\\left(\\frac{-3z^2-4}{p}\\right)\\) roots in \\(x\\). The elementary identity\n\\[\n\\sum_{z\\in\\mathbf F_p}\\left(\\frac{az^2+c}{p}\\right)=-\\left(\\frac a p\\right)\\qquad(ac\\ne0)\n\\]\nshows that the conic has \\(p-\\chi\\) ordered points. To prove the identity, factor out \\(a\\) and use \\(\\sum_z\\left(\\frac{z^2-t}{p}\\right)=-1\\) for \\(t\\ne0\\). The latter sum counts the \\(p-1\\) factorizations \\((z-u)(z+u)=t\\).\n\nThe diagonal contributes \\(1+\\chi\\) points. Hence the number of ordered collisions with \\(x\\ne z\\) is \\(p-1-2\\chi\\). It also equals \\(2N_2+6N_3\\), giving\n\\[\nN_1=\\frac{p-1}{2},\\qquad N_2=1+\\chi,\\qquad N_3=\\frac{p-3-4\\chi}{6}.\n\\]\nThe image size is \\(p-N_2-2N_3=(2p+\\chi)/3\\). This derivation separates the repeated-root fibers that a three-to-one heuristic misses.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "the fibers of x cubed plus x over every prime field of characteristic greater than 3"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1016/j.jnt.2024.04.016",
      "locator": "Self-contained collision derivation recorded here; its final value-set formula agrees with Brazelton-Harrington-Litman-Wong, arXiv:2103.09119v2, page 1, and the classical references cited there"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/j.jnt.2024.04.016",
    "locator": "Self-contained collision derivation recorded here; its final value-set formula agrees with Brazelton-Harrington-Litman-Wong, arXiv:2103.09119v2, page 1, and the classical references cited there"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R154",
      "title": "The image of x cubed plus x has the proposed size",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R152",
      "title": "Exact image enumeration for 5,131 primes",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "cubic-map-image-size",
      "title": "cubic map image size",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}