{
  "schema_version": 1,
  "problem_number": "OWR-13750332-001",
  "title": "Star Transforms Without Type 2 Singular Directions and Conflitti's Conjecture on Elementary Symmetric Polynomials",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "The star transform integrates a function on the plane along m rays with a common vertex, with directions γ_1, …, γ_m and nonzero weights c_1, …, c_m. In the inversion formula of Ambartsoumian and Latifi, the directions ψ at which Σ_j c_j Π_{i≠j} ⟨ψ, γ_i⟩ vanishes, called singular directions of Type 2, cause instability. They showed that every star with an even number of rays has such directions, and that for m = 3 every choice of weights admits ray directions without them. For odd m ≥ 5 this was left as a conjecture, which the Oberwolfach Report 21/2023 records as unproven. We prove the conjecture for every odd m and all nonzero real weights. The proof reduces the problem, by nearly parallel rays, to a rational function of one variable without real zeros, which we build by nesting explicit three-pole clusters. The same report recalls a conjecture of Conflitti (2006): for even r, the real zero set of the elementary symmetric polynomial e_r in n variables contains no linear subspace of dimension r. Previously the cases r = 2 (Conflitti) and n = r + 1 (Ambartsoumian and Latifi; this case also follows from the irreducibility of e_{n−1}) were known. We prove the conjecture for all even r and all n, using Descartes' rule of signs and a parity argument. Hence the largest subspace in the zero set has dimension min(n, r − 1). For even r ≥ 4 we also show that the only (r − 1)-dimensional subspaces in the zero set are coordinate subspaces. Exact computer certificates for explicit stars with at most 21 rays illustrate the construction; the proofs do not depend on them. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AG",
    "math.CA",
    "math.FA"
  ],
  "keywords": [
    "star transform",
    "singular directions of Type 2",
    "stable inversion",
    "tomography",
    "elementary symmetric polynomial",
    "real zero set",
    "linear subspaces",
    "Descartes' rule of signs",
    "Conflitti's conjecture",
    "Oberwolfach Reports",
    "OWR-13750332-001",
    "math.AG",
    "math.CA",
    "math.FA",
    "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-13750332-001/",
  "pdf_url": "https://eulersolve.org/papers/owr-13750332-001/paper.pdf?v=8dd18022e8c5",
  "doi": "10.5281/zenodo.23062557",
  "zenodo_record_url": "https://zenodo.org/records/23062557",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Proves the Ambartsoumian–Latifi conjecture (OWR 21/2023) for every odd number of rays and all nonzero real weights, and Conflitti's conjecture for all even r and all n. The star construction proves existence without a quantitative stability bound.",
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    "source.zip": {
      "sha256": "1ffd59f14a34a73b6c9358b2c05e7578828618fcfcc775b6586083442cad5565"
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    "verification_report.md": {
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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."
}