{
  "schema_version": 1,
  "problem_number": "AMR-021-0006",
  "title": "Sharp Finite Zero Bounds for One-Parameter Separated Polynomial Systems",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let k >= 2 be even. We determine the largest finite real solution set of a separated system f_i(x_i) = g_i(t), 1 <= i <= m, with all degrees at most k. The maximum is k^2/4 for m = 1, and m(k-1)k^m/4 for m >= 2. The upper bound follows from a weighted count of isolated feasible points of univariate polynomial inequalities: root multiplicity pays for the loss of choices in a fiber at a vanishing constraint. The proof allows constant right-hand sides, multiple roots, and a positive-dimensional complex zero set. For m >= 2, the matching construction is the previously published even-degree family of DannyExperiments (DOI 10.5281/zenodo.21875290); the contribution here is the upper bound throughout the separated class. The unrestricted extremal problem for isolated real zeros of sums of squares is not resolved. The supplement contains exact Python and C++ checks. This is an unrefereed preprint, not a formal certification or an absolute priority claim.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.AG",
    "math.CO"
  ],
  "keywords": [
    "real algebraic geometry",
    "separated polynomial system",
    "sums of squares",
    "finite real zero set",
    "root multiplicity",
    "AMR-021-0006"
  ],
  "manuscript_version_date": "2026-10-17",
  "publication_date": "2026-10-17",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-17",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-021-0006/",
  "pdf_url": "https://eulersolve.org/papers/amr-021-0006/paper.pdf?v=38a2c4fca716",
  "doi": "10.5281/zenodo.23438315",
  "zenodo_record_url": "https://zenodo.org/records/23438315",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The exact maximum is proved only for one-parameter separated systems with an even degree cap and a finite entire real solution set. The unrestricted Shapiro Problem 3 remains open. DannyExperiments' attaining family is credited prior work; the contribution is the restricted upper bound. AI-assisted, self-audited and unrefereed; no independent human review, formal verification or certified absolute priority.",
  "files": {
    "paper.pdf": {
      "sha256": "38a2c4fca71605feabb6878e176702cfdfa9c18b0bdabc785a4bbe490ce8e4ab"
    },
    "source.zip": {
      "sha256": "7d3d36dc6fcaa5a84f328affb6311f1ff69b73157ee9c81dc18bc8909aa1eb1c"
    },
    "verification_report.md": {
      "sha256": "48e9fedb66861d9eca7fb98cc1c91bbada6887d6575bb0b35f22617b9a1e99b3"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}