# Verification report — OWR-9790358-016 (algebraic relations among mean value coordinates)

Verification date: 2026-09-30. Two independent AI-assisted verification runs checked the note; this revision
applies the changes requested by the second run.

**Verdict.** This is a scoped partial answer. It is complete for quadrilaterals and gives structure for every n.
- **Quadrilaterals with no three vertices collinear** (convex, non-convex or self-intersecting): the question is answered
  completely. The ideal of the Zariski closure S_P of the mean value image is principal, I(S_P) = (F_P). Here F_P is an
  explicit irreducible form of degree 14: F_P = N_4(w_1²R_1, …, w_4²R_4)/s², the rationalized form of the relation
  λ_1|x−v_1| − λ_2|x−v_2| + λ_3|x−v_3| − λ_4|x−v_4| = 0.
- **Every n ≥ 4, proved:**
  - S_P is an irreducible surface;
  - the projection w ↦ Σw_iv_i/Σw_i restricts to a map of degree 2^{n−1} from S_P to the plane;
  - off the centre Λ = {s = m = 0}, the fibres consist of the "mean value weights with signed distances", and they are
    characterized by circumscribed polygons.
- **n = 5, 6, 7:** computations modulo primes for sample polygons, which are listed in `reproducibility/`.
  - For four hexagons it is certified that I(S_P) has nothing below degree 6 and that I(S_P)_6 is spanned by a
    Brianchon sextic.
  - The relations of lowest degree 10 (n = 5) and 7 (n = 7) rest on quotients computed modulo p.
  - deg S_P = 152 comes from a single hexagon.
- **Open:** a generating set for n ≥ 5, and the degree formula 2^{n−3}(6n−17), which is only conjectural.

The note is unrefereed.

## Statement checked
- **Primary source.** Mini-Workshop "Interpolation, Approximation, and Algebra", Oberwolfach Reports 19 (2022),
  no. 1, 383–428, Report No. 7/2022, doi:10.4171/OWR/2022/7 (organizers C. de Boor, T. Sauer, H. K. Schenck,
  T. Sorokina).
  - The item is "Mean-value coordinates" (Sottile) in the problem session on p. 424. The report text was read.
  - Sottile asks: "What are the homogeneous equations for the Zariski closure of the image?"
  - The image is that of an n-gon under its mean value coordinates, in P^{n−1} or in the affine plane Σλ = 1.
- **Corpus record.** ulamai/UnsolvedMath, OWR-9790358-016 (status `partially_solved`), titled "What about in higher
  dimensions?". Its statement consists of three pieces from p. 424:
  - the question "What about in higher dimensions?" from the item "Three dimensional Wachspress varieties"
    (Sottile, Schenck);
  - the whole mean value item;
  - the whole item "Moduli spaces of Wachspress varieties" (Sottile).

  The current label rests on Wachspress literature (Kohn–Ranestad). The note treats only the mean value item.

## Readings
| Reading | Answered? | Where |
|---|---|---|
| n = 3 | yes, trivially: S_P = P², no equations | §2 |
| n = 4, no three vertices collinear, any cyclic order | yes, completely: I(S_P) = (F_P), F_P irreducible of degree 14; p_0 has multiplicity 6 | Thm 1.1, Cor. 5.3 |
| general n: description of the image closure | yes, structure proved (fibres, degree 2^{n−1}, circumscribed-polygon criterion off Λ, rationalized Pitot relation for even n) | Thm 1.2, Thm 3.2, Prop. 4.2 |
| general n: a generating set of I(S_P) | no, open; data for n = 5, 6, 7 mod p for sample polygons, and a certified lowest-degree statement for four hexagons | §6 |
| quadrilaterals with three collinear vertices (degenerate) | not covered by Thm 1.1, and the hypothesis is needed: for two examples F_P = w_j²K with deg K = 12 and I(S_P) = (K) (computational) | Remark 5.6 |

## Results in the paper
- **Lemma 2.1** (three-point form, as in Floater–Hormann–Kós 2006): w = M(x)r. M(x) is symmetric of rank n−2, its image
  is K(x) = {Σw_i(v_i−x) = 0}, and s(M(x)ρ) = Σκ_iρ_i/(A_{i−1}A_i).
- **Lemmas 2.2–2.3.** The cover X = {ρ_i² = |v_i−x|²} is irreducible (Kummer theory). S_P = closure of Φ(X°) is an
  irreducible surface, independent of the open part of the polygon used.
- **The projection π = [s : m_1 : m_2] and its centre Λ** (§1 and §2). They are the tautological linear projection and
  its centre of projection in Garcia-Puente–Sottile 2010, and the linear projection of Irving's thesis (2012).
  - Wachspress coordinates: π is birational on the Wachspress surface W. For a quadrilateral, W is a quadric in P³ and
    the centre is a point on W.
  - Mean value coordinates: π has degree 2^{n−1} on S_P. For a quadrilateral, S_P has degree 14 and p_0 has
    multiplicity 6.
- **Theorem 3.2 (all n ≥ 4).** For x in a dense open set V:
  - the 2^{n−1} points [M(x)(εr)] are distinct;
  - they lie off Λ, and they are exactly S_P ∩ π^{-1}(x);
  - a point w ∈ K(x) with Σw ≠ 0 lies on S_P iff the side lines of the polygon with edge vectors w_iJ(v_i−x) are
    tangent to a common circle.

  Distinctness is proved through the Kummer-basis expansion of det[ερ, ε'ρ, D¹, D²] (coefficient ±2 det(v_a−x, v_b−x)).
  Consequences: deg(π|S_P) = 2^{n−1}, and X/±1 → S_P is birational. Remark 3.3 discusses the centre Λ and the cone
  point p_0.
- **Proposition 3.4** (Pitot relations; immediate from Floater's definition): w_iρ_i = t_{i−1} + t_i on every
  branch, and Σ(−1)^i w_iρ_i = 0 for even n.
- **Lemma 4.1 and Proposition 4.2.** s² divides N_P = N_n(w_i²R_i). For even n, F_P = N_P/s² is a nonzero element of
  I(S_P) of degree 2^n − 2.
- **Lemma 5.1.** 2F_P ≡ (m·m)² C(w) ∏_{η≠1}(η·w) mod s, with C(w) = Σ_i w_i det(v_i, m)². So s does not divide F_P.
- **Lemma 5.2.** Some factor of F_P vanishes identically on K(x) only for at most 7 points x.
- **Theorem 1.1 / proof in §5.** I(S_P) = (F_P). The argument uses:
  - simple zeros of F_P on the lines K(x) at the 8 fibre points;
  - a cone argument for any extra factor;
  - exclusion of cone factors by Lemmas 5.1 and 5.2.
- **Corollary 5.3.** p_0 has multiplicity 6. The tangent cone is the cone over the projective plane sextic
  T = N_4(k_i²R_i)/s² = 0; in the chart s = 1 its equation is N_4(k_i² q_i(x)) = 0.
  - The paragraph after the proof shows that s divides T when v_av_b ∥ v_cv_d for a partition {a,b} ∪ {c,d}.
  - Exact computation gives T = s²·(quartic) for the unit square and the other parallelograms tested, and
    T = s·(quintic) for the trapezoids tested.
  - In these cases the affine equation has degree less than 6 and the tangent cone contains the plane s = 0.
- **Remark 5.4.** F_P does not depend on the cyclic order, so the three quadrilaterals on four points share S_P.
- **Example 5.5.** The unit square: F_P/16 has 116 terms, coefficients ≤ 27 in absolute value, 21 dihedral orbits
  (Table 1).
- **Remark 5.6** (three collinear vertices; computational). If three vertices lie on a line, Lemma 5.2 fails and F_P
  vanishes on the plane w_j = 0 (v_j the fourth vertex). For two examples:
  - F_P = w_j²K exactly, with deg K = 12;
  - E_11 has full rank mod p, and dim ker E_d = 1, 4, 10 for d = 12, 13, 14;
  - hence I(S_P) = (K), deg S_P = 12, and F_P does not generate I(S_P).
- **§6 (computations for sample polygons, labelled as such).**
  - Lemma 6.1: kernel dimensions of mod-p evaluation matrices are upper bounds for dim I(S_P)_d, and full rank is a
    certificate.
  - Tangency conditions: circle conditions (n = 5), the Brianchon sextic (n = 6, certified for four hexagons), conic
    conditions (n = 7). The relations of degree 10 (n = 5) and 7 (n = 7) rest on quotients computed modulo p.
  - Table 2 gives dimensions and generator counts. I(S_P)_d = 0 for d < d_0 is certified by full rank for the
    polygons tested.
  - deg S_P = 52 (n = 5) and 152 (n = 6, one hexagon); deg C_L = 14, 38, 94.
  - Conjecture 6.2: deg S_P = 2^{n−3}(6n−17).

## Computations (scripts and outputs in reproducibility/)
- **Author** (`lead/`).
  - Exact symbolic identities over Z (`lead_identities.py`), with negative controls.
  - Exact integer checks for 31 quadrilaterals (`lead_quad_instances.py`): s² | N_P, Lemma 5.1, order 6 and tangent
    cone at p_0, Proposition 4.2(a), and F_P(W(x,ρ)) = 0 on all 16 branches at once.
  - The unit-square table (`lead_square_orbits.py`).
  - The modular certificate for n = 4 (`lead_modp_certificate.py`): rank E_13 = 560 (full) and rank E_14 = 679.
  - Lowest degrees for n = 5, 6, 7 on six polygons (`lead_low_degree.py`).
  - The exact Brianchon certificate for four hexagons (`lead_hexagon_brianchon.py`): Br = s⁶K_6, K_6 has 50 terms,
    rank E_5 = 252, rank E_6 = 461.
  - Added in this revision (`lead_quad_extras.py`):
    - the exact power of s dividing T (Corollary 5.3) for 12 quadrilaterals; in each case it equals the number of
      parallel pairs v_av_b ∥ v_cv_d;
    - the factorization F_P = w_j²K and the ranks of E_11, …, E_14 for the two examples of Remark 5.6.
- **Finder** (`claimant/`).
  - Exact F_P for two quadrilaterals.
  - Mod-p checks of Lemma 3.1, Proposition 3.4 and Theorem 3.2(c) for 13 polygons (n = 4..7, every sign vector).
  - Modular ranks for 10 quadrilaterals.
  - All n ≥ 5 computations: Hilbert-function ranks, generator counts, tangency conditions, curve degrees.
- **First independent verification runs** (`verifier/`, AI-assisted, code written before the finder's programs were
  read; outputs rerun on 2026-09-30).
  - Float checks.
  - Exact F_P of the square.
  - Kernel dimensions:
    - n = 4: 0 up to d = 13 and 1 at d = 14, for five quadrilaterals;
    - n = 5: 0, 2, 16 at d = 9, 10, 11;
    - n = 6: 0, 1 at d = 5, 6;
    - n = 7: 0, 8 at d = 6, 7.
  - Circle and Brianchon conditions.
  - deg C_L = 14, 38, 94.
  - deg S_P = 52 for a further pentagon (nullities 0, 1, 3 in degrees 51, 52, 53).
  - Nonzero collision determinants.

  Not reproduced independently: deg S_6 = 152, and the generator counts beyond degree 11.
- **Second independent verification run** (`independent_run_2/`, AI-assisted, new code). Its points of S_P over F_p
  come from Floater's definition. For the release, comments and seed labels were given neutral wording and the
  programs were rerun on 2026-09-30; all counts agree with the original run.
  - Exact checks for 16 quadrilaterals:
    - s² | N_P, deg F_P = 14, Lemma 5.1, k_i ≠ 0;
    - order 6 at p_0, with lowest part N_4(k_i²R_i)/s²;
    - F_P(w) = 0 on all 16 branches, with w from the definition;
    - Remark 5.4, and Example 5.5 / Table 1.
  - Ranks for 12 quadrilaterals at two primes: dim ker E_d = 0, 0, 1, 4, 10 for d = 12..16. With the exact F_P this
    gives I(S_P) = (F_P). A negative control is included.
  - General n:
    - Lemma 2.1(b),(c) exactly for n = 4..8;
    - definition = three-point form and the Pitot relation in the Kummer algebra (n ≤ 6);
    - Theorem 3.2(a),(c) mod p for n = 4..8;
    - the collision determinant for all 28 pairs of sign classes;
    - the key step of the proof of Theorem 1.1;
    - the example of Remark 3.3.
  - Real-number checks with 60 to 150 digits: F_P(λ) = 0 at real points, the product formula after Theorem 1.1,
    Lemma 4.1 for n = 5, 6, 7, and Proposition 4.2(a) for n = 5, 6.
  - The hexagon certificate for the four hexagons of the paper and two random ones.
  - Table 2 kernel dimensions on polygons of its own:
    - n = 5: 0, 2, 16, 59, 150, 309 for d = 9..14;
    - n = 6: 21, 56, 129, 345 for d = 8..11;
    - n = 7: 0, 8, 49 for d = 6, 7, 8.
  - Corollary 5.3: the power of s dividing T for five quadrilaterals.
  - The two three-collinear examples of Remark 5.6 (dim ker E_12,13,14 = 1, 4, 10).

  Not reproduced by this run: deg S_6 = 152, the generator counts, the n = 5 circle-condition quotients, the n = 7
  conic conditions, and deg C_L.

## Independent verification

### First independent verification run
Verdicts of the first independent verification run (2026-09-30):

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (mean value item only; the record merges three items) |
| Proofs | CONFIRMED after the required fixes (formula (*), linear precision, circumscribed-polygon identity and converse, Pitot, and the six steps of Theorem 1.1, each checked step by step in the AI-assisted verification run) |
| Computations | CONFIRMED for n = 4 and the low-degree data for n = 5, 6, 7; deg S_6 = 152 not reproduced |
| Answer as posed | PARTIAL: complete for n = 4, structure for all n, generators for n ≥ 5 open |
| Novelty | apparently new; no priority claim |

All five required fixes were applied:
1. **Distinct fibre points.** Theorem 3.2(a) now contains a proof that the fibre points over a general x are distinct.
   It uses the Kummer-basis expansion of det[ερ, ε'ρ, D¹, D²], whose coefficient of ρ_jρ_k is ±2 det(v_a − x, v_b − x).
   The proof holds for every quadrilateral with distinct vertices, and in fact for every n ≥ 4. It replaces the
   earlier numerical check.
2. **The centre Λ.** Theorem 3.2(b),(c) and Theorem 1.2(b) are restricted to points off Λ = {s = m = 0}.
   Remark 3.3 explains why: for n = 4, p_0 ∈ S_P and every line P(K(x)) passes through p_0.
3. **Credit.**
   - The three-point form is due to Floater–Hormann–Kós.
   - w_i r_i = t_{i−1} + t_i is Floater's definition, so the Pitot relation is immediate.
   - The direction from coordinates to circumscribed polygons is the classical tangent-length and polar-dual picture
     (Floater 2003; Ju–Schaefer–Warren–Desbrun 2005).
   - Floater–Muntingh 2025 is cited.
   - Only the converse, I(S_P) = (F_P) for n = 4, and the n ≥ 5 structure and data are presented as new.
4. **Labels for n ≥ 5.** Every n ≥ 5 statement is labelled as a computation modulo p. Kernel dimensions are upper
   bounds and full-rank results are certificates (Lemma 6.1). deg S_6 = 152 is labelled computed, and
   2^{n−3}(6n−17) conjectural.
5. **Scope.** The note is restricted to the mean value item, and it suggests splitting the record.

Further changes in that revision:
- an exact certificate for four hexagons;
- exact symbolic checks of the polynomial identities used in the proofs (Lemma 5.1 is checked on exact instances);
- 31 exact quadrilateral instances;
- the remark that F_P does not depend on the cyclic order.

The claim that S_P meets Λ for n ≥ 5 is stated only as a computational indication.

### Second independent verification run
A second independent verification run, also AI-assisted, checked the revised note on 2026-09-30.
- It checked the proofs step by step and recomputed the results with new code (`independent_run_2/`).
- It reran the released programs from `source.zip` and found the recorded outputs reproduced.
- It made new anonymous literature searches.
- It found no mathematical error and no prior publication.

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (the question is quoted exactly; only the mean value item of the merged record is treated) |
| Proofs | CORRECT, including the counting argument in the proof of Theorem 1.1; one imprecise sentence in Corollary 5.3 |
| Computations | REPRODUCED with new code for n = 4 and for the kernel dimensions of Table 2; the released programs reproduce their recorded outputs; deg S_6 = 152 and the generator counts not reproduced |
| Novelty and credit | apparently new; two credits were missing |
| Presentation | good; scope qualifiers for n ≥ 5, a notation clash, and the wording of one row of this report |
| Mathematical error | none |

Its five required changes are applied in this revision:
1. **Credit.** Two works are now cited.
   - Garcia-Puente–Sottile, "Linear precision for parametric patches", Adv. Comput. Math. 33 (2010), no. 2, 191–214,
     doi:10.1007/s10444-009-9126-7 (checked on Crossref; arXiv:0706.2116 read).
   - Irving, "Wachspress Varieties", PhD thesis, Texas A&M University, December 2012, read in the copy at
     https://franksottile.github.io/advising/irving.pdf.

   The sentence "We did not see Irving's thesis" is replaced. §1 and §2 identify π and Λ with the linear projection
   and centre of projection of these works, and contrast the two cases, as summarized under "Results in the paper".
   Remark 3.3 notes that for Wachspress surfaces the centre meets W in the image of the adjoint curve.
2. **This report.** The row "Proofs" of the first run's table did not describe the check of the six steps of
   Theorem 1.1 as part of an AI-assisted verification run. It now says that they were checked step by step in the
   AI-assisted verification run. The Zenodo copy of this report, `source.zip` and the checksums were regenerated.
3. **Scope qualifiers for n ≥ 5.**
   - The Table 2 caption says that I(S_P)_d = 0 for d < d_0 is certified "for the polygons tested (listed in the
     release)".
   - §1 and §6 say "for the sample polygons". They state that the relations of degree 10 (n = 5) and 7 (n = 7) rest
     on quotients computed modulo p, whereas d_0 = 6 is certified for four hexagons.
   - The abstract and the Zenodo description say "computations modulo primes for sample polygons", and that
     deg S_P = 152 comes from a single hexagon.
4. **Notation.** The Table 2 caption writes C[w]_1 · ker E_{d−1}, with C[w]_1 the linear forms, instead of
   R_1 · ker E_{d−1}; R_1 is the quadric of (2).
5. **Corollary 5.3.** The tangent cone is now given as the cone over the projective plane sextic
   T = N_4(k_i²R_i)/s² = 0, with affine form N_4(k_i²q_i(x)) in the chart s = 1.
   - A new paragraph proves that s divides T when two of the lines v_av_b, v_cv_d are parallel.
   - It records T = s²·(quartic) for the unit square and T = s·(quintic) for the trapezoids tested, checked by an exact
     computation (`lead/lead_quad_extras.py`).
   - In these cases the affine equation has degree less than 6 and the tangent cone contains the plane s = 0.

Optional suggestions of the second run:
- Applied:
  - the precise description of the merged corpus record (§1 and "Scope and priority");
  - the remark on three collinear vertices (Remark 5.6, labelled computational);
  - the second run is recorded in the Verification paragraph of the paper.
- Not applied:
  - an exact division over Z for the n = 5 circle conditions, which would be a new large computation;
  - the citation of Dieci–Difonzo (arXiv:2208.01037);
  - a wording change in the proof of Lemma 2.3(b);
  - a relabelled line in a recorded output of the first run, which is kept as it was run.
- Already in the paper: the Hilbert-function data for n = 5 in the paragraph "Degrees" of §6.

Other changes in this revision:
- Stale theorem numbers in the docstrings of four programs in `lead/` were updated. Their outputs are unchanged.
- The second run's programs and outputs were added as `reproducibility/independent_run_2/`.

## Relation to the literature, novelty and scope
- **Searches (September 2026).**
  - arXiv API: "mean value coordinates" with algebraic, variety, Zariski, implicit or Wachspress; Wachspress titles;
    "barycentric" with Zariski closure or algebraic relations; au:Sottile; tangential and circumscribed polygons;
    Pitot.
  - Crossref, OpenAlex (partly rate-limited) and zbMATH.
  - Three single web searches in total.
  - The second verification run searched again:
    - 9 arXiv queries;
    - Crossref for every DOI of the bibliography;
    - the DOI handle API;
    - 44 zbMATH documents on "mean value coordinates";
    - one web search.

    OpenAlex was unavailable. It read Floater–Muntingh (arXiv:2407.00422) and Irving's thesis in full.
  - For this revision: the Crossref record of Garcia-Puente–Sottile, its arXiv version and Irving's thesis (anonymous
    requests).

  None gives implicit equations, degrees, ideals or the circumscribed-polygon characterization of mean value images.
- **Closest work.**
  - Floater 2003 and Hormann–Floater 2006 (definition); Floater–Hormann–Kós 2006 (three-point form).
  - Ju–Schaefer–Warren–Desbrun 2005 (polar duals); Floater 2015 (survey).
  - Floater–Kosinka 2010 and Floater–Muntingh 2025 (injectivity; the latter uses half-angle-tangent identities for
    quadrilaterals, not an implicit equation).
  - Garcia-Puente–Sottile 2010: linear precision and the tautological linear projection, which is birational on the
    image closure exactly when the patch has rational linear precision.
  - Irving 2012: Wachspress varieties studied through the same projection.
  - Irving–Schenck 2014 and Kohn–Ranestad 2020 (Wachspress surfaces and varieties).
- **DOIs.** All were checked through the Crossref API. The exception is Ju–Schaefer–Warren–Desbrun
  (10.2312/SGP/SGP05/181-186), which is not in Crossref and was checked through the DOI handle system. Irving's thesis
  has no DOI and is cited by its URL.
- **Caveats.** This negative search is not a proof of priority.
- **Status.** The merged record stays `partially_solved`. For the mean value item alone the honest status is
  "partially solved": complete for n = 4 with no three vertices collinear, open for n ≥ 5.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/