{
  "schema_version": 1,
  "problem_number": "OWR-12697711-015",
  "title": "The Amoeba Dimension of an Arbitrary Loopless Matroid",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let M be a loopless matroid on a finite set E with rank function r, and let Σ(M) ⊆ ℝ^E be the support of its matroid fan. Draisma, Eggleston, Pendavingh, Rau and Yuen defined adim(M) as the minimum of 2 dim(Σ(M) + R) − dim R over rational subspaces R ⊆ ℝ^E; for the matroid of a complex linear space this is the dimension of the amoeba of the space. They proved that adim(M) = min Σ_i (2r(P_i) − 1), the minimum over all partitions {P_1, …, P_k} of E, when M is realizable over ℂ, and asked whether this holds for every loopless matroid. The question appears in an Oberwolfach report, in a BIRS problem session and as Conjecture 1.4.1 of their paper. We show that the answer is yes, and that both numbers equal dim(Σ(M) + Σ(M)); the minimum is attained by a rational subspace in the braid arrangement. The lower bound dim(Σ(M) + Σ(M)) ≥ min Σ_i (2r(P_i) − 1) is a theorem of Bernstein, valid for all loopless matroids. The only new ingredient is the inequality 2 dim(Φ + R) − dim R ≥ dim(Φ + Φ) for every finite union of cones Φ and every subspace R, which follows from the Grassmann formula. Consequently S ↦ adim(M|S) is a matroid rank function, which answers one of the sub-questions of the BIRS problem. The others are answered by earlier results: the partition minimum for M|S is a matroid rank function of S (Draisma et al.), and by Bernstein's theorem it equals dim(Σ(M|S) + Σ(M|S)). The inequality can be strict for other fans. Exact computations on connected non-realizable matroids are consistent with the theorem. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.AG"
  ],
  "keywords": [
    "amoeba dimension",
    "matroid",
    "matroid fan",
    "Bergman fan",
    "Minkowski sum",
    "Dilworth truncation",
    "non-realizable matroids",
    "tropical geometry",
    "Oberwolfach Reports",
    "OWR-12697711-015",
    "math.CO",
    "math.AG"
  ],
  "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-12697711-015/",
  "pdf_url": "https://eulersolve.org/papers/owr-12697711-015/paper.pdf?v=bf4587c3571b",
  "doi": "10.5281/zenodo.23058144",
  "zenodo_record_url": "https://zenodo.org/records/23058144",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers the arbitrary-loopless-matroid amoeba-dimension question. Bernstein's theorem supplies the lower bound for all loopless matroids; the new ingredient is the cone-family inequality from the Grassmann formula. The restriction-rank sub-question follows, while the other BIRS sub-questions are answered by earlier results of Draisma et al. and Bernstein. Exact computations are consistency checks. AI-assisted, self-audited and unrefereed; no independent human peer review is claimed.",
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    "source.zip": {
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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."
}