{
  "schema_version": 1,
  "problem_number": "OWR-9790358-009",
  "title": "Universal Polynomial Interpolation Need Not Be Hereditary",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "A finite family of affine algebraic sets may allow polynomial interpolation of every pointwise-compatible tuple although a subfamily does not. For every integer m >= 2, this note constructs an irredundant family of three proper, reduced, pure curves in C^3 with this property, while the sum of two vanishing ideals has quotient C[x]/(x^m). An explicit formula interpolates all compatible data on the whole family. The construction disproves the necessity direction of the radical-chain criterion in Theorem 2.5 of McKinley, Shekhtman and Tuesink, arXiv:2208.12406v1, even after correcting its indexing. The note gives the precise hereditary replacement and proves that the full-family criterion is restored when all pairwise intersections are finite. Classical two-set interpolation and Chinese remainder methods are credited. This is an unrefereed preprint, not a claim to a new solution of the full general interpolation program or an absolute priority claim.",
  "result_type": "COMPLETE_SCOPED_COUNTEREXAMPLE_AND_REPAIR",
  "categories": [
    "math.AC",
    "math.AG"
  ],
  "keywords": [
    "OWR-9790358-009",
    "math.AC",
    "math.AG"
  ],
  "manuscript_version_date": "2026-10-21",
  "publication_date": "2026-10-23",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-23",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-9790358-009/",
  "pdf_url": "https://eulersolve.org/papers/owr-9790358-009/paper.pdf?v=61462fe6c423",
  "doi": "10.5281/zenodo.23586340",
  "zenodo_record_url": "https://zenodo.org/records/23586340",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every m>=2, an irredundant family of three proper reduced pure curves in C^3 admits interpolation of every pointwise-compatible tuple although the first two vanishing ideals sum to (x^m,y,z). This disproves necessity in the corrected radical-chain criterion of arXiv:2208.12406v1 Theorem 2.5 and OWR 7/2022 Theorem 4. The all-subfamily radicality criterion is exactly hereditary interpolation; finite pairwise intersections restore the full-family equivalence. The broader frozen question and PDE constraints are not closed, and classical CRT methods are not claimed as new. This is a self-audited, AI-assisted, unrefereed preprint. No independent certification, formal proof verification or absolute-priority claim is made.",
  "files": {
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      "sha256": "61462fe6c4239f6705e1e5aa0a6b35562cf80f9103252f932f0989e824851fdd"
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    "source.zip": {
      "sha256": "e26d4cca07824badd91932ea4b8191bd3316ad763197c2b65e3b53dbc6986424"
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    "verification_report.md": {
      "sha256": "301545591ac7aa0e52bfaecd69124876f453cd2718c139b8b7b4eb06971c43d3"
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  "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."
}