Published paper
An Identity for the Coefficients of Characteristic Polynomials of Hyperplane Arrangements
MSC 2020
52C3551M2052A5551M0452A2260D0552B1151F15
Abstract
For a finite affine hyperplane arrangement, the paper proves that, outside a finite union of affine hyperplanes, the number of chambers whose metric projection contains the projecting point in a face of dimension is the absolute value of the -th characteristic-polynomial coefficient.
Role in dependence graphs
Proof-critical source
Interpreting the (signed) chromatic polynomial coefficients via hyperplane arrangements
This paper is included only for the following marked statement:
- Theorem 1.2 · journal p. 1479; full proof pp. 1487–1496Exact affine-arrangement projection-coefficient theorem imported as focal Theorem 2.5.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Open version of record · Discrete & Computational Geometry 70 (2023), 1476–1498
Zakhar Kabluchko. An Identity for the Coefficients of Characteristic Polynomials of Hyperplane Arrangements. Discrete & Computational Geometry 70 (2023), 1476–1498.
Open audited source ↗01Statements2 reported findingsCorrect
The paper's projection-count theorem, explicit exceptional-set theorem, reflection-arrangement corollary, and worked arrangement examples are correct with their stated hypotheses. No false or unsupported mathematical statement was found in the complete version of record.
Theorems 1.2 and 1.6Correct
The projection statistic equals the characteristic coefficient
Journal pages 1479–1482 · Theorems 1.2 and 1.6
For an affine arrangement in dimension , the number of closed chambers whose metric projection of a generic point lies in a -face is the absolute -th characteristic coefficient. Theorem 1.6 makes the exceptional set explicit as a finite union of proper affine hyperplanes. The indexing, signs, and affine rather than merely central scope agree throughout the statement and proof.
Exact open version of record ↗Examples 1.3–1.4 and Section 4Correct
The main specializations and examples are consistent
Journal pages 1479–1481 and 1494–1497 · Examples 1.3–1.4 and Section 4
Summing the dimension-refined counts recovers Zaslavsky's region formula, while the group action converts the chamber count to the Drton–Klivans orbit count. The Boolean and coordinate examples have the expected characteristic coefficients and projection dimensions. No boundary case contradicts the closed-chamber convention stated on journal page 1478.
02Proofs3 reported findingsCorrect
The complete proof chain was checked. Normal-cone polarity translates projection faces into chamber incidences, the localized central-arrangement count supplies the coefficient contribution, and the finite exceptional family makes every required incidence locally constant. The final Whitney-formula calculation has the claimed coefficient and sign.
Proposition 2.1 and Theorem 1.6Correct
The metric-projection criterion is applied with the correct face dimension
Journal pages 1482–1490 · Proposition 2.1 and proof of Theorem 1.6, Steps 1–2
For a face of a chamber , the inverse image of under metric projection is . Passing to the polar of the tangent cone gives the needed chamber-intersection problem in the orthogonal complement. Relative interiors are used at the exact points needed to avoid double counting lower-dimensional faces.
Proposition 3.1 and proof Steps 3–4Correct
The local chamber count and global coefficient sum close the argument
Journal pages 1485–1496 · Proposition 3.1 and proof of Theorem 1.6, Steps 3–4
The central localization at an intersection flat has the required general-position section after the exceptional hyperplanes are removed. The polar-cone count is then summed over flats and chambers. Reordering that finite sum produces the Whitney expansion of the characteristic polynomial, with the parity absorbed by the nonnegative coefficient convention. The cited KVZ chamber-section result is checked separately in its exact version of record.
Exceptional setCorrect
The genericity set is genuinely a finite hyperplane union
Journal pages 1480–1482 and 1488–1492 · definition of and proof Steps 1–3
Only finitely many arrangement faces, their spans, and their normal-cone boundary spans occur. Every excluded condition is contained in a proper affine hyperplane. Thus the theorem proves the stronger advertised conclusion rather than only an unspecified null-set statement.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.