# Verification report — OWR-12697711-015 (amoeba dimensions of arbitrary loopless matroids) Verification date: 2026-09-30. **Verdict.** The answer is yes. For every loopless matroid M on a finite set E, adim(M) = dim(Σ(M) + Σ(M)) = r'(E) = min over partitions {P_1, …, P_k} of E of Σ_i (2 r(P_i) − 1), and the minimum defining adim(M) is attained by the rational braid-arrangement subspace R_P of any optimal partition P. This proves Conjecture 1.4.1 of Draisma–Eggleston–Pendavingh–Rau–Yuen and answers Open question 2 of the Oberwolfach abstract and problem 3.7.1 of the BIRS 23w5149 report. Of the BIRS sub-questions, the note newly answers whether f2(S) = adim(M|S) is a matroid rank function (yes, since f2 = f3 = r'); the other sub-questions are answered by earlier results (DEPRY Theorem 1.3.2; Bernstein's Theorem 3.5 with Edmonds' rank formula). The published literature lists the case of matroids not realizable over ℂ as open; the affirmative answer in that case rests on this note. The lower bound is Bernstein's theorem (2022); the only new ingredient is an elementary inequality (Lemma 3.1). The note is unrefereed. ## Statement checked - **Primary source.** J. Draisma (joint work with S. Eggleston, R. Pendavingh, J. Rau and C. H. Yuen), "Amoeba dimensions", Oberwolfach Reports 20 (2023), no. 1, pp. 854–857, in Report 15/2023 "New Directions in Real Algebraic Geometry", doi:10.4171/OWR/2023/15. - Open question 2 is on p. 857: for an arbitrary loopless matroid M on [n] with matroid-fan support F, is min{2 dim(U + F) − dim U : U spanned by rational vectors} equal to min Σ_i (2 rk_M(P_i) − 1) over partitions of [n] into nonempty parts? - The abstract notes that "≤" is easy (U spanned by the indicator vectors of the parts) and says of the converse: "For the converse we have no idea!" - **Same question as a conjecture.** J. Draisma, S. Eggleston, R. Pendavingh, J. Rau, C. H. Yuen, "The amoeba dimension of a linear space", Proc. Amer. Math. Soc. 152 (2024), no. 6, 2385–2401, doi:10.1090/proc/16744 (read in arXiv:2303.13143v3). Conjecture 1.4.1: the minimum in (3) is attained in the braid arrangement. It is proved there for matroids realizable over ℂ (Theorem 1.3.1 with the Draisma–Rau–Yuen formula). - **The two forms.** DEPRY §1.4 state that the conjecture is equivalent to r'(E) = adim(M). Their argument uses the identity r~(P) = 2 dim(Σ(M) + R_P) − dim R_P for every partition P, which holds in general only as the inequality ≤ (see Lemma 3.3 below). - **What holds a priori.** By Lemma 3.3, adim(M) ≤ min_P (2 dim(Σ(M) + R_P) − dim R_P) ≤ r'(E), so r'(E) = adim(M) implies the braid form. The converse needs 2 dim(Σ(M) + R_P) − dim R_P ≥ r'(E) for every partition P, which is part of Theorem 1.2 (Corollary 3.2 with Theorem 3.4 and Lemma 3.7). - **In the paper.** Problem 1.1 states the OWR form (a) and the DEPRY form (b) separately, and Theorem 1.2 proves both. - **Third place it was posed.** M. Baker, J. Huh, F. Rincón, K. Shaw, "Algebraic Aspects of Matroid Theory (23w5149)", BIRS workshop report, 2023, §3.7, problem 1 (J. Rau). It asks whether the two minima are always equal and, for the set functions f1(S) = r(S), f2(S) = adim(M|S), f3(S) = r'(S): is f2 a matroid rank function? f3? What is an interpretation of f3? - **Corpus record.** ulamai/UnsolvedMath, OWR-12697711-015 (status `partially_solved`; assessment "the non-realizable case remains conjectural"). Minor record issues: 1. The `statement.original` field is garbled by PDF extraction in the partition formula ("k X G min{ (2 rkM (Pi ) − 1) | Pi = [n] and all Pi 6= ∅}"); the clean statement is correct. 2. The OWR abstract's reference [1] prints the arXiv number of DEPRY as 2023.13143; the correct number is 2303.13143. 3. The literature assessment was accurate for the published literature; it becomes outdated only through this note. ## Readings | Reading | Answer | Where | |---|---|---| | minimum over rational subspaces (OWR; DEPRY (3)) | yes, = r'(E) | Theorem 1.2 | | minimum over all real subspaces | yes, same value | Theorem 1.2 (Corollary 3.2 holds for every real R) | | DEPRY Conjecture 1.4.1: the minimum is attained in the braid arrangement | yes, at R_P for every optimal P | Theorem 1.2 | | equivalence of the OWR form (a) and the DEPRY form (b) | (a) ⇒ (b) a priori (Lemma 3.3); (b) ⇒ (a) needs the lower bound of Theorem 1.2; both forms hold | Problem 1.1 and the paragraph after it | | sign convention of the matroid (Bergman) fan | irrelevant (all quantities are invariant under Σ ↦ −Σ) | §2 | | BIRS sub-questions: f2 = f3? f2, f3 rank functions? interpretation of f3? | yes (new); yes (f2: new; f3: already DEPRY Thm 1.3.2); f3(S) = dim(Σ(M\|S) + Σ(M\|S)), which already follows from Bernstein's Thm 3.5 with Edmonds' rank formula, since π_S(Σ(M)) = Σ(M\|S) | Corollary 4.1 and the paragraph after it | | two matroids M, N | min_R [dim(Σ(M)+R) + dim(Σ(N)+R) − dim R] = dim(Σ(M)+Σ(N)) = min_P Σ (r_M(P_i) + r_N(P_i) − 1) | Proposition 4.2 | | arbitrary rational fans (OWR Open question 1, separate record OWR-12697711-014) | not settled; the inequality of Corollary 3.2 can be strict | Proposition 5.1, Remark 5.2 | ## Results in the paper - **Lemma 3.1 (new).** dim(A + B) + dim R ≤ dim(A + R) + dim(B + R) for subspaces A, B, R (Grassmann formula). **Corollary 3.2.** 2 dim(Φ + R) − dim R ≥ dim(Φ + Φ) for every finite union of convex cones Φ and every subspace R; hence adim(M) ≥ dim(Σ(M) + Σ(M)). - **Lemma 3.3 (OWR; DEPRY §1.4).** 2 dim(Σ(M) + R_P) − dim R_P ≤ Σ_i (2 r(P_i) − 1); hence adim(M) ≤ r'(E). - The inequality can be strict. Since 1_E lies in L_F and in R_P, dim(L_F + R_P) ≤ d + k − 1, so dim(Σ(M) + R_P) ≤ min{Σ_i r(P_i), d + k − 1}. Examples: U_{2,4} with P = {{1,2},{3,4}} gives 4 < 6; F7 with P = {two points | the other five} gives 6 < 8. - Equality holds for every optimal partition, by Theorem 1.2. - So the per-partition identity stated in DEPRY §1.4 holds in general only as the inequality ≤. Apart from their remark on the equivalence of the two forms, that inequality is all that DEPRY use. - The pre-revision version of the paper said, citing DEPRY §1.4, that equality always holds. That sentence was false, although nothing used it; it was replaced by the paragraph after Lemma 3.3. - **Theorem 3.4 (Bernstein, SIAM J. Appl. Algebra Geom. 6 (2022), Lemma 3.4 and Theorem 3.5).** If |J| ≤ r_M(J) + r_N(J) − 1 for all nonempty J ⊆ I, then I is a forest in H_{F,G} for suitable maximal flags, so dim(Σ(M) + Σ(N)) ≥ |I|. Bernstein stated and proved this for all loopless matroids; the paper includes a proof for completeness (Lemma 3.5 lifting flags from a restriction; Lemma 3.6 a greedy procedure). - **Lemma 3.7.** A basis I of the DEPRY matroid M′ has |I| = r'(E) and is 2-sparse (DEPRY Theorem 1.3.2, or Edmonds' rank formula for induced matroids). - **Theorem 1.2 (main).** Combination of the three ingredients above. - **Corollary 4.1.** f2(S) = adim(M|S) equals f3(S) = r'(S), and f3(S) = dim(Σ(M|S) + Σ(M|S)). - The only new part is f2 = f3, which shows that f2 is a matroid rank function. - That f3 is a rank function is DEPRY Theorem 1.3.2. - The description f3(S) = dim(Σ(M|S) + Σ(M|S)) is Bernstein's: by Edmonds' rank formula f3 is the rank function of the matroid induced by 2r − 1; by Bernstein's Theorem 3.5 its rank at S is dim π_S(Σ(M) + Σ(M)); and π_S(Σ(M)) = Σ(M|S) (chains of flats of M|S, and Lemma 3.5). - **Proposition 4.2, Remark 4.3.** Two-matroid version; for realizable M, another proof of DEPRY Theorem 1.3.1 (tropical counterpart of Antolini–Dewar–Tanigawa §6). - **Proposition 5.1.** For the union Φ of the four coordinate 3-planes of ℝ^6 indexed by the vertex stars of K4 (pairwise meeting in a line), dim(Φ + Φ) = 5 but min_R (2 dim(Φ + R) − dim R) = 6. This fan is balanced but not connected in codimension one, so it lies outside the class of Draisma–Rau–Yuen's Question 14; whether the inequality can be strict within that class is left open. - **Remark 5.2.** For X ⊆ (ℂ*)^n irreducible, dim_ℝ A(X) = dim(Trop X + Trop X), with a proof; its literature status was not checked, and it is not used elsewhere. ## Computations (exact; scripts and outputs in reproducibility/) All computations are consistency checks; no proof depends on them. - **Claimant** (`claimant/`, standard library only). - `verify_all.py`: 35 matroids (F7, F7*, AG(3,2), Vámos, Vámos*, Pappus, non-Pappus, non-Pappus*, Desargues, non-Desargues, F7 ⊕ U(2,3), and random sparse paving, dual and rational matroids) and 10 pairs: ALL OK (45/45). - `exhaustive_check.py`: all 5596 labelled sparse paving matroids of rank 4 on 7 elements; 3000 random of rank 4 on 8 and 800 of rank 5 on 9; 1500 random pairs on 8 elements: failures = 0. - `certificates.py`: explicit flags with dim(L_F + L_G) = r'(E) for F7*, Vámos, non-Pappus*, AG(3,2), F7. - Reruns on 2026-09-30 reproduce the recorded outputs apart from timings. - **Verifier** (`verifier/indep_verify.py`, written from the paper's text without the claimant's library). - Part 1: 22 pairwise non-isomorphic connected matroids that contain F7, AG(3,2), Vámos, non-Pappus or non-Desargues as a restriction (so they are not realizable over ℂ) and have r'(E) < min(n, 2d − 1) (Table 1 of the paper; n ≤ 13). For each: exhaustive dim(Σ + Σ) = r'(E) = f(R_P), a forest certificate with exact rank, and f(R) ≥ r'(E) on 24 random rational R. - Part 1b: 16 further connected free extensions of F7, F7*, AG(3,2), Vámos and non-Pappus. - Part 2: 11 named, realizable or disconnected matroids. - Part 3: Proposition 5.1 on all 64 coordinate subspaces and 3000 random rational subspaces. - Result: ALL OK (about 4 minutes). - **Further independent verification run** (`independent_run_2/`, written from the definitions and the text of the note before the programs above were read; standard library only; exact ranks). - `run2_check.py` (two random seeds): 10 named matroids; the 22 matroids of Table 1; 8 further matroids not realizable over ℂ; 115 random sparse paving matroids; 3 matroids realizable over ℚ; 18 pairs for Proposition 4.2; Lemma 3.6 on 300 random instances; Proposition 5.1 on 4267 subspaces; the value at R_P for every partition P of 60 matroids with n ≤ 8. Result: ALL CHECKS PASSED (about 1 minute per seed). - `run2_extra1.py` to `run2_extra5.py`: the U_{2,4} example; distinct rank–size profiles of the Table 1 matroids and dim π_S(Σ+Σ) = dim(Σ(M|S)+Σ(M|S)) = r'(S) for 1787 sets S; all 27 labelled loopless matroids on 4 elements against all 1084 subspaces of ℚ^4 spanned by {−1,0,1}-vectors; all 21147 partitions of six matroids on 9 elements; three further non-realizable matroids. No failures. - `rerun_of_release_scripts/`: the run's reruns of the four release programs, identical to the recorded outputs apart from timings. - **Coverage note.** In the claimant's single-matroid and exhaustive test sets, 34 of 35 matroids and all 9396 matroids have r'(E) = min(n, 2d − 1), i.e. a trivial optimal partition; the only exception is the disconnected F7 ⊕ U(2,3). The verifier's Part 1 covers the connected non-realizable regime with a nontrivial optimal partition. ## Independent verification First independent verification run (2026-09-30; AI-assisted): - Classification: paper candidate. Correct: yes. Answers the question as intended: yes. - Source: the OWR 15/2023 PDF was fetched anonymously; Open question 2 is on p. 857 and matches the record. - Proofs: every step checked (fan structure, rank of the incidence matrix, Lemma A, the use of DEPRY Theorem 1.3.2, the lifting lemma, the greedy proof of Bernstein's lemma, the upper bound). The greedy proof agrees with Bernstein's Lemma 3.4 (arXiv v4), whose Theorem 3.5 is stated for arbitrary loopless matroids. - Computations: the claimant's programs were re-run with identical results apart from timings; separate scratch code tested 195 matroids and 54 pairs (including 22 connected non-ℂ-realizable matroids with a nontrivial optimal partition) with no failures. That scratch code was not preserved; `verifier/indep_verify.py` repeats these checks. - Novelty: nothing in print resolves the question (see below). All seven required fixes were applied: 1. Publication before any status change: this package is prepared for a DOI deposit; the corpus status should change only after the note has a DOI, and the HF notice should cite it and say that the old assessment was accurate for the published literature. 2. Credit: the lower bound is attributed to Bernstein (Theorem 3.4), who covered all loopless matroids; only Lemma 3.1 is claimed as new; the folklore caveat (in ADT, Antolini thanks Eggleston) is in the Scope paragraph. 3. The BIRS 23w5149 report is cited as a third source, and its sub-questions are discussed (Corollary 4.1; the credit for the interpretation of f3 was corrected after the further run below). 4. The proof structure is described consistently as three ingredients. 5. Connected non-ℂ-realizable test cases with a nontrivial optimal partition were added (Table 1). 6. The strictness example for general fans is Proposition 5.1; no status change carries over to OWR-12697711-014, and the identity of Remark 5.2 is marked as not literature-checked. 7. The record corrections (garbled `statement.original`; arXiv number misprint in the OWR abstract) are kept. Further independent verification run (2026-09-30, after the revision above; AI-assisted; programs and outputs in `reproducibility/independent_run_2/`): - Verdict: correct; minor revision; nothing fatal. Main theorem, lemmas, Proposition 4.2, Proposition 5.1 and Remark 5.2 checked step by step. - Sources: OWR 15/2023 (EMS Press PDF; Open question 2 on p. 857), DEPRY arXiv:2303.13143v3 (latest version per the arXiv API), Bernstein arXiv:2003.10529v4, the BIRS 23w5149 report, DRY20 (arXiv:1812.08149v2), Nisse–Sottile "Describing amoebas" (arXiv:1805.00273v3) and ADT (arXiv:2508.04798v1), all fetched anonymously. Statement fidelity confirmed. - Findings (all addressed in this revision): 1. False side remark after Lemma 3.3: "equality holds in the second claim [DEPRY §1.4]". It fails in general (U_{2,4} with P = {{1,2},{3,4}}: 4 < 6; F7 with P = {two points | the other five}: 6 < 8; strict inequality for most partitions in full scans). Replaced by a correct paragraph (bound min{Σ r(P_i), d + k − 1}; equality for optimal partitions by Theorem 1.2; the DEPRY identity holds in general only as ≤, which is all DEPRY use apart from their equivalence remark). 2. The "Equivalently" in Problem 1.1 relied on that identity. Problem 1.1 now states the OWR form (a) and the DEPRY form (b) separately, with the correct justification ((a) ⇒ (b) by Lemma 3.3; (b) ⇒ (a) only through the lower bound of Theorem 1.2); Theorem 1.2 proves both. The matching line of this report was corrected. 3. Credit for the third BIRS sub-question: f3(S) = dim(Σ(M|S) + Σ(M|S)) already follows from Bernstein's Theorem 3.5 with Edmonds' rank formula, because π_S(Σ(M)) = Σ(M|S). The paper now says so, states that the only new part of Corollary 4.1 is f2 = f3, and the abstract was reworded accordingly. 4. Release regenerated (paper, source archive, Zenodo copies, checksums); this report updated. - Optional suggestions applied: DRY20 Corollary 7 cited for formula (1) and Question 14 discussed after Proposition 5.1; Φ instead of Σ in the abstract for a general union of cones; Table 1 kept out of the Verification list; "by hand" dropped; the ADT acknowledgement attributed to Antolini alone; in Remark 5.2 the semicontinuity step now refers to smooth points of X × X̄, and Trop(X × X̄) = Trop X × Trop X̄ is stated before applying Sturmfels–Tevelev; the run's programs added to the release (`reproducibility/independent_run_2/`). - Computations: see "Further independent verification run" under Computations. The release programs were rerun from the extracted source archive with identical results apart from timings. - Novelty: no prior resolution found (searches below). ## Relation to the literature, novelty and scope - **Searches (September 2026).** - arXiv API: "amoeba dimension"; amoeba with matroid; amoebas with matroids; amoeba with Bergman; "Bergman fan(s)" with Minkowski; "matroid fan"; "Dilworth truncation"; "Hadamard product" with matroid; amoeba with tropical and linear; amoeba with dimension; author listings of the DEPRY authors and of Bernstein. - OpenAlex and OpenCitations: works citing DEPRY (only Antolini–Dewar–Tanigawa) and Bernstein 2022 (none other on amoebas); zbMATH (DEPRY review Zbl 1547.14080; ADT entry); Crossref (publication data, DOIs). - One web search in each of two runs (only DEPRY, ADT and the BIRS report came up), and one in the further run (DEPRY, ADT, arXiv:2511.22646 and arXiv:2207.13639 came up; none resolves Conjecture 1.4.1). - Further run: arXiv API topic queries and author listings (Draisma, Eggleston, Pendavingh, Rau, Yuen, Bernstein, Nisse, Sottile); OpenCitations (DEPRY is cited only by ADT; Bernstein 2022 by 9 works, none other on amoebas; DRY20 by Nisse–Sottile 2022, a survey by Krasikov 2023 and DEPRY); zbMATH (no documents citing DEPRY; DRY20 cited only by DEPRY); Crossref (all DOIs of the paper). OpenAlex refused anonymous requests (HTTP 429), so the cached responses of the first search were used. - Full texts read: the OWR abstract, the BIRS report, DEPRY (arXiv v3), Bernstein (arXiv v4), ADT (arXiv v1 and the published abstract), and arXiv:2511.22646, which quotes Bernstein's theorem for general matroids and does not mention amoebas. The further run also read DRY20 (arXiv:1812.08149v2: Corollary 7 is formula (1); Question 14 is still open) and Nisse–Sottile, "Describing amoebas" (arXiv:1805.00273v3); neither contains the lower bound adim(M) ≥ dim(Σ(M) + Σ(M)) or the identity of Remark 5.2. - **Findings.** No source resolves Open question 2 / Conjecture 1.4.1 for non-realizable matroids. ADT (SIAM J. Discrete Math. 40 (2026)) re-prove the DEPRY formula for complex linear spaces only. - **Credit.** Lower bound: Bernstein 2022. Upper bound: OWR and DEPRY. The matroid M′: DEPRY Theorem 1.3.2. Induced matroids and Dilworth truncation: Edmonds; ADT Theorem 2.7. The description f3(S) = r'(S) = dim(Σ(M|S) + Σ(M|S)): Bernstein's Theorem 3.5 with Edmonds' rank formula. New: Lemma 3.1 only (and, through it, f2 = f3). - **Caveats.** The argument is short and uses only published ingredients, so it may be known informally (in ADT, Antolini thanks Eggleston, a DEPRY author, for initiating the discussion on amoebas of linear spaces). This negative search is not a proof of priority. The identity of Remark 5.2 was not literature-checked. The published PAMS version of DEPRY was not read; its bibliographic data were confirmed via zbMATH. - **Scope.** The note settles OWR-12697711-015 for all loopless matroids. It does not settle OWR-12697711-014 (an algorithm for arbitrary rational fans). ## 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/