{
  "schema_version": 1,
  "problem_number": "OWR-9790358-016",
  "title": "Algebraic Relations Among Mean Value Coordinates: A Complete Answer for Quadrilaterals and Partial Results for Larger Polygons",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Mean value coordinates of a planar polygon P with n vertices define a map from P to P^(n−1). At an Oberwolfach mini-workshop in 2022, F. Sottile asked for the homogeneous equations of the Zariski closure S_P of its image. We answer this question for quadrilaterals and give partial results for larger polygons. For every quadrilateral with no three vertices collinear, the ideal of S_P is generated by one explicit irreducible polynomial F_P of degree 14. It is obtained by clearing the square roots in the relation λ_1|x − v_1| − λ_2|x − v_2| + λ_3|x − v_3| − λ_4|x − v_4| = 0, which follows at once from Floater's formula; for the unit square, F_P/16 has 116 integer terms. For every n ≥ 4, S_P is an irreducible surface, and the projection w ↦ Σ w_i v_i / Σ w_i restricts to a map of degree 2^(n−1) from S_P to the plane. Over a general point x, a point w with Σ w_i ≠ 0 and Σ w_i (v_i − x) = 0 lies on S_P exactly when the closed polygon with edge vectors w_i J(v_i − x), J a quarter turn, is circumscribed about a circle. The passage from coordinates to circumscribed polygons is the classical tangent-length picture behind Floater's construction; its converse is what produces equations. For n = 5, 6, 7, computations modulo primes for sample polygons give relations of lowest degree 10, 6, 7, and deg S_P = 52 for n = 5; for a single hexagon they give deg S_P = 152. These values are consistent with a conjectured value 2^(n−3)(6n − 17). A generating set for n ≥ 5 remains open. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.AG",
    "math.NA"
  ],
  "keywords": [
    "mean value coordinates",
    "generalized barycentric coordinates",
    "Zariski closure",
    "implicit equation",
    "algebraic relations",
    "tangential polygon",
    "Pitot theorem",
    "Brianchon theorem",
    "quadrilateral",
    "Oberwolfach Reports",
    "OWR-9790358-016",
    "math.AG",
    "math.NA",
    "cs.CG",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-9790358-016/",
  "pdf_url": "https://eulersolve.org/papers/owr-9790358-016/paper.pdf?v=09c2cbbf8c15",
  "doi": "10.5281/zenodo.23064820",
  "zenodo_record_url": "https://zenodo.org/records/23064820",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Sottile's question (OWR 7/2022, p. 424) completely for quadrilaterals with no three vertices collinear (a degree-14 generator); for n = 5, 6, 7 only computations modulo primes for sample polygons are reported, and a generating set for n ≥ 5 remains open. The HF record merges three OWR items; only the mean value item is addressed.",
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    "source.zip": {
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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."
}