# Verification report — AMR-014-0004 (Fröberg–Lundqvist–Oneto–Shapiro, Conjecture 1.9: maximal k-rank of binary forms) Verification date: 2026-10-14. **Verdict.** The note gives a **counterexample**: Conjecture 1.9 of the source is false. Every statement it makes is proved; two steps are computer-assisted, as listed below. - **Refuted.** For every d ≥ 2 and every k ∈ {5, 6, 7, 8, 9}, and every odd k with 7 ≤ k ≤ 201, the form x·y^{kd−1} is a sum of fewer than k k-th powers of binary forms of degree d (4, 5, 6, 6, 7 terms for k = 5, …, 9; k − 1 terms for the odd k). So the conjunction (A) ∧ (B) of the conjecture fails at all these (k, d), and clause (B) fails in the reading rk_k(ℓ1ℓ2^{kd−1}) = k of Lundqvist–Oneto–Reznick–Shapiro. - **No computer needed** for k = 7: x·y^13 is an explicit sum of six 7th powers of binary quadratic forms (Theorem 1.2). - **Computer-assisted.** (C1)–(C3) for 7 ≤ k ≤ 201 (Theorem 1.3; finite integer computations); the certified identities for k = 5, 6, 8, 9 (and a second one for k = 7) (Theorem 1.4; interval/Newton–Kantorovich certificates, checked by six independent programs). - **Known before (credited).** The conjecture holds for k ≤ 2, for d = 1, and at (k, d) = (3,2), (3,3), (3,4), (4,2) (FLOS; Reznick; LORS; Carlini–Oneto; Bik–Draisma–Oneto–Ventura). Hence (5,2) is the first failure in the order of k for d = 2. The construction of Theorems 1.2 and 1.3 is a variant of Reznick's "synching" of powers of quadratic trinomials (Pacific J. Math. 303 (2019), Theorem 2.1 and Corollary 2.3); the identities themselves are new as far as the searches show. - **Open.** Clause (A) alone outside the known cases; k = 4 with d ≥ 3; even k ≥ 10 and odd k > 201; the exact values of rk_k(x·y^{kd−1}) for k ≥ 5. - **Consequences.** Problem D of the source (record AMR-014-0006) has a negative answer; upper bounds below N for Heilbronn's problem (entire functions) and its analogues for polynomials (the Waring number of C[t]), rational and meromorphic functions, for N = 5, …, 9 and odd 7 ≤ N ≤ 201 (novelty not established: two primary sources unread). The note is unrefereed. ## Statement checked - **Primary source.** R. Fröberg, S. Lundqvist, A. Oneto, B. Shapiro, "Algebraic stories from one and from the other pockets", Arnold Math. J. 4 (2018) 137–160, doi:10.1007/s40598-018-0088-z; arXiv:1801.01692 (v1, the only version). - Read in the arXiv TeX source (fetched 2026-10-14) and compared with the journal version: Definition 1.5 (k-rank), Definition 1.8 (maximal k-rank; p. 141), Conjecture 1.9 (p. 142), equation (5) (the case k = 2; p. 142), Remark 1.10 (Blekherman–Teitler), Problem D (k-ranks of binary monomials; p. 143). Problems are lettered. - Conjecture 1.9: for all positive integers k, d, (A) rk^max_k(kd, 2) = k, and (B) the forms ℓ1ℓ2^{kd−1} (ℓ1, ℓ2 non-proportional linear forms) have this maximal k-rank. - Definition 1.8 literally says "in n+1 variables", while the rest of the section uses n variables, Conjecture 1.9 speaks of binary forms, and the case k = 2 is proved by factoring a binary form. The literal (ternary) reading would make the conjecture false at (k, d) = (2, 1); the binary reading is the intended one (as in LORS). - **Neighbour source.** S. Lundqvist, A. Oneto, B. Reznick, B. Shapiro, "On generic and maximal k-ranks of binary forms", J. Pure Appl. Algebra 223 (2019) 2062–2079 (arXiv:1711.05014v1 read): Conjecture 1.4 (clause (A), Shapiro, 2014); the sentence "One can also suspect that rk^max_k(l1 l2^{kd−1}) = k" (rk^max applied to a form, i.e. rk_k), the source of clause (B); Theorem 2.3 (generic k-rank ⌈(kd+1)/(d+1)⌉), Theorem 3.1 (binary sextics), Example 4.7 (rk_4(xy^7) = 4, Macaulay2), Proposition B.1 (rk_k(m) ≤ k for (k−2)n ≤ d; its introduction and FLOS say "smaller than k" / "less than k", a slip for "at most k"). - **Corpus record.** ulamai/UnsolvedMath AMR-014-0004 (dataset version 1.6.0; upstream status `partially_solved`; "source-order Conjecture 4"). Its statement is Conjecture 1.9 verbatim. ## Readings | Reading | Answer | Where | |---|---|---| | Conjecture 1.9 as the conjunction (A) ∧ (B), for all k, d | false at (k, d) for k = 5, …, 9 and odd 7 ≤ k ≤ 201, all d ≥ 2 | Corollary 1.5 | | (B) in the LORS reading rk_k(ℓ1ℓ2^{kd−1}) = k | false at the same (k, d) | Corollary 1.5 | | (B) in the FLOS reading rk_k(ℓ1ℓ2^{kd−1}) = rk^max_k(kd, 2) | false wherever (A) holds; which clause fails is not decided | Section 1.3 | | (A) alone (Shapiro's conjecture) | not decided at any new (k, d); known at k ≤ 2, d = 1, (3,2), (3,3), (3,4), (4,2) | Sections 1.2, 7 | | Literal "n+1 variables" (ternary forms) | not meant; would fail already at (2, 1) | Section 1.1 | | Real forms | not the question (the source is over C) | — | | "At most m" vs "exactly m" terms; forms divisible by y allowed | immaterial | Section 1.1 | ## Results in the paper - **Lemma 2.1.** GL2-invariance; x·y^{kd−1} = Σ G_i^k (G_i of degree d) iff t = Σ g_i(t)^k (deg g_i ≤ d); lift from d = 2 to all d ≥ 2 by multiplying by y^{d−2}; the case split showing that rk_k(x·y^{kd−1}) < k refutes (A) ∧ (B). - **Formulas (3)–(5), Lemma 2.2.** Coefficients of (1 + bt + ct²)^k, the reflection rule Q_{2k−n} = c^{k−n}Q_n, the polynomials φ_n, averaging over roots of unity. - **Theorem 1.2 (k = 7).** s² = 165, I∓ = (−15 ∓ s)/10, β² = I_−, q³ = −(15053 − 420s)/14053, p² = I_+q, h = 1 + βt + t², g = 1 + pt + qt², A = −β/(pq⁶), σ = 7β(1 − q^−6): t = (1/(3σ)) Σ_{j<3} ω^{−j}(h(ω^j t)^7 + A g(ω^j t)^7); so x·y^13 is a sum of six 7th powers of binary quadratic forms. Proof by multinomial coefficients and the cube-root-of-unity average; Table 1 gives the six forms numerically. - **Proposition 4.1, Lemma 4.2, Theorem 1.3.** The same construction for odd k = 2m + 1 ≥ 7 gives k − 1 terms whenever (C1) φ_{m+1} is squarefree, (C2) φ_{m+1}, φ_k have no common root, (C3) φ_k² mod φ_{m+1} is not constant; a modular criterion for (C1)–(C3); checked for all 98 odd k in [7, 201]. Example 4.3: k = 9 in closed form (√2, ω = i). - **Lemma 5.1, Theorem 1.4.** A Krawczyk-type existence and uniqueness test with proof; six certificates (60-digit centres, radius 2^−100) for (k, s) = (5,4), (6,5) [two], (7,6), (8,6), (9,7); Table 3. - **Corollary 1.5, Propositions 6.1, 6.2.** The refutation for all d ≥ 2; first failure (5,2) in the order of k for d = 2; Problem D negative; g_C(N) ≤ 4, 5, 6, 6, 7 for N = 5, …, 9 and g_C(N) ≤ N − 1 for odd 7 ≤ N ≤ 201 in each of the classes P, E, R, M of Hayman–Lingham's Problem 2.26. ## Computations (programs and outputs in reproducibility/) - **Programs of the note.** `check_note.py` (standard library, 4.4 s; ALL CHECKS PASSED): formulas (3)–(5) for k ≤ 21; Theorem 1.2 and Example 4.3 by exact normal forms (15 and 19 coefficients); Lemma 4.2(ii) modulo 2^61 − 1 for all 98 odd k ≤ 201 and (C1)–(C3) exactly for k ≤ 61; the six certificates with exact rational majorants (ρ between 1.0·10^−14 and 3.5·10^−12; bounds for |φ(X) − c⁰| between 1.2·10^−44 and 4.2·10^−42, r = 7.9·10^−31); 18 corrupted certificates, all failing; numerical sanity checks (48 root choices for k = 7, coefficient error ≤ 4.5·10^−14). `rejection_tests_k7_exact.py` (SymPy, 1 s): four corrupted k = 7 identities, all rejected; it regenerates the recorded output of the first stage byte for byte (the original script had not been kept). - **Programs of the first stage** (`original/`). Checker 1 (integer interval arithmetic on the grid 2^−512, F(c⁰) exact) and checker 2 (mpmath interval arithmetic, 400 bits, real form): all six certificates PASS, also at 256 / 250 bits; 36 corruptions rejected by both. `check_k7_exact.py` (SymPy): identity (7) exact. `exact_symmetric.py 201`: (C1)–(C3) exactly over Q for all odd k ≤ 201 (35–44 s). Exact residuals at the centres between 3.5·10^−60 and 1.6·10^−58. - **Re-runs.** On 2026-10-14 all programs with recorded outputs were run again from a copy of the package: 23 quick outputs byte-identical (`run_quick.sh`), all slow outputs identical up to running times (`RERUN_LOG.txt`). The run on the final text did the same again from an extracted copy of the archive, with the same result (`independent_run_2/r2_package_rerun.txt`). ## Independent verification runs The results were first obtained, with proofs and programs, in the morning of 2026-10-14. Three independent verification runs, all AI-assisted, followed; each wrote its own programs. Runs A and B worked separately on the results before the note was written; the third run examined the final text of the note and the package (below). | Item | Run A | Run B | |---|---|---| | Statement, reading, logic of the refutation (case split (A)/(B)), GL2, dehomogenisation, lift to d ≥ 2 | CONFIRMED (re-derived) | CONFIRMED (re-derived) | | Theorem 1.2 (k = 7, closed form) | CONFIRMED: proof re-derived; own exact arithmetic in Λ; five corruptions rejected; 100 digits, 24 root choices (7.4·10^−92) | CONFIRMED: proof re-derived; own exact arithmetic; five corruptions rejected; explicit 7th roots, 110 digits, 48 root choices (5.4·10^−109) | | Proposition 4.1 and (C1)–(C3) for 7 ≤ k ≤ 201 | CONFIRMED: construction re-derived; exact and modulo 2^61 − 1 (φ_n by expansion), 98/98; numerical identities up to k = 101 | CONFIRMED: construction re-derived; modulo two primes, 98/98; exact for k ≤ 61; numerical identities k ≤ 21; also 203 ≤ k ≤ 1001 (not used) | | Lemma 5.1 and the checkers | CONFIRMED | CONFIRMED (checkers read line by line; no error) | | Certificates k = 5, …, 9 (Theorem 1.4) | CONFIRMED by its own exact-majorant checker (6 PASS; 35 corrupted certificates rejected: seven corruptions of each of five certificates, 56 tests with repetitions); original checkers re-run (36/36 rejected) | CONFIRMED by its own Newton–Kantorovich checker (6 PASS; 42 corruptions rejected); own search and own certificates for k = 5 (three orbits), 6, 8 | | Known cases, "first failure" | CONFIRMED_WITH_FIXES: add BDOV 2020; credit Carlini–Oneto for rk_3 = 3 and x⁶y² | CONFIRMED_WITH_FIXES: add BDOV 2020; "first" only in the order of k for d = 2 | | Problem D (AMR-014-0006) negative | CONFIRMED | CONFIRMED | | Heilbronn bounds | CONFIRMED as bounds; novelty hedged (wording fixes) | CONFIRMED as bounds; novelty not established (NS79, Hayman 1985 unread) | | Overall | CONFIRMED_WITH_FIXES | CONFIRMED (with four required fixes) | Neither run found an error in the identities, the certificates, Lemma 5.1, the checkers, the construction or the reduction. ### Third run: on the final text (`reproducibility/independent_run_2/`) The third run, also AI-assisted, examined the final text of the note and the package line by line, as if nothing had been checked, with programs of its own (FLINT/Arb through python-flint 0.7.1, and mpmath); verdict **CONFIRMED WITH FIXES**, no mathematical error. - **Sources.** It read again, in the versions named in the note, FLOS §1 (with the journal pages), LORS §1–4 and App. B, Carlini–Oneto §1–3, BDOV §1 (Theorem 1.6), Hayman–Lingham Problem 2.26 and Update, Voloch 2015 completely, and found and read Reznick, *Linearly dependent powers of binary quadratic forms* (Pacific J. Math. 303 (2019) 729–755; arXiv v1, Sections 1, 2, 5). Every cited statement agrees with these texts; the record's statement is Conjecture 1.9 verbatim. - **Proofs.** Lemma 2.1, formulas (3)–(5), Lemma 2.2, Theorem 1.2 (every number of Steps 1–4), Proposition 4.1, Lemma 4.2, Example 4.3, Remark 4.4, Lemma 5.1, Remark 5.2, Corollary 1.5, Propositions 6.1 and 6.2 and the bounds of Sections 6.2 and 7: re-derived; correct. - **Own computations.** `r2_k7_closed_form.py`: Steps 1–2 exactly in Q(√165); identity (7) in its own tower arithmetic for Λ (15 coefficients zero; a corrupted version rejected); the six-term identity in Arb at 3000 bits for all 48 root choices (error ≤ 1.6·10^−897); Table 1 reproduced. `r2_oddk.py`: for all 98 odd k in [7, 201], φ_n by expansion, (C1)–(C3) exactly over Q, Lemma 4.2(ii) modulo 10^9 + 7 and 998244353, the hypothesis of Proposition 4.1 by rigorous root isolation, and identity (8) numerically for k ≤ 101: all pass (28 s). `r2_krawczyk_arb.py`: Lemma 5.1 for the six certificates in Arb ball arithmetic: all PASS (row sums ≤ 6.0·10^−25, bounds (b) ≤ 8.5·10^−25 r, residuals as in Table 3, |c_i2| ≥ 0.108 on every box); 36 corrupted certificates rejected; Remark 5.2 at the centre (8.4·10^−61). `r2_small_claims.py`: s(k) = ⌈(2k+2)/3⌉, the parameter count, Problem D, the bounds of Section 6.2, the printed k = 5 identity (16 digits), and that the k = 7 certificate is not in the orbit of Theorem 1.2 (orbit invariants differ). - **Package.** Re-run from an extracted copy of `source.zip`: 23/23 quick outputs byte-identical, 14 hashes OK, all slow outputs of `original/`, `verification_run_A/` and `verification_run_B/` identical or identical up to running times. - **Literature.** arXiv API, OpenAlex citing lists (FLOS, LORS, Carlini–Oneto, Newman–Slater), zbMATH, Crossref and one web search: no refutation of the conjecture and no representation of x·y^{2k−1} with fewer than k terms. **Corrections required by the runs**, all applied in the note: 1. (A, B) Cite Bik–Draisma–Oneto–Ventura, *The monic rank*, Math. Comp. 89 (2020), Theorem 1.6: clause (A) at (3,3), (3,4), (4,2); with the known lower bounds the whole conjecture holds at (3,2), (3,3), (3,4), (4,2); "(4, d) for d ≥ 3 is open"; "(5,2) is the first failure" only in the order of k for d = 2. — Sections 1.2, 1.3, 6, 7, abstract. 2. (A, B) Quote Hayman–Lingham with "for large k" and "substantially better"; year 2019 (Springer book). — Section 6.2. 3. (A, B) Add Voloch 2015 (√n < v(0, n) ≤ n, quoted from Newman–Slater; his question v(0, n) = n/2 + O(1)); keep every novelty statement on Heilbronn's problem hedged; say that Newman–Slater 1979 and Hayman 1985 were not read. — Sections 6.2, 8. 4. (A) Range "odd k with 7 ≤ k ≤ 201" everywhere. — Abstract, Theorem 1.3, Corollary 1.5, Proposition 6.2. 5. (A) Quote the LORS sentence exactly (with rk^max applied to a form). — Section 1.1. 6. (A) "Waring problem for polynomials" in FLOS means forms; the relevant problem is called "Heilbronn's problem / the Waring number of C[t]". — Sections 1.3, 6.2, 8. 7. (A) Credit rk_3(x·y^{3d−1}) = 3 and the 4-rank 3 of x⁶y² to Carlini–Oneto. — Sections 1.2, 6.1, 8. 8. (A) Reading of (B): positive statements about (B) at k = 3 hold in the LORS reading. — Section 1.3 (only the cases where (A) is known are claimed). 9. (B) Add the script behind `rejection_tests_k7_exact.txt` and document `show_identity.py 22`. — `reproducibility/rejection_tests_k7_exact.py`, `reproducibility/README.md`, Section 8. 10. (third run) Credit Reznick's synching (Pacific J. Math. 303 (2019) 729–755, Theorem 2.1 and Corollary 2.3 of arXiv v1; doi:10.2140/pjm.2019.303.729, checked with Crossref): averaging powers of quadratic trinomials over roots of unity with the middle coefficient chosen as a root of a coefficient polynomial is the technique of Theorem 1.2 and Proposition 4.1. — Abstract, Sections 1.4, 4, 8 ("Scope and priority", "What was read"), bibliography. 11. (third run) Heilbronn's problem concerns entire functions; its polynomial analogue is the Waring number of C[t]. — Abstract, Section 1.3. 12. (third run) Numbers of run A: 35 distinct corrupted certificates (not 56), |φ(X) − c⁰| ≤ 2.6·10^−54 (the recorded maximum is 2.53·10^−54). — Section 8 and this report. 13. (third run) "Open otherwise" for clause (A) is a statement about the literature: "as far as we know". — Section 7. 14. (third run) Record the third run. — Sections 3, 4, 5, 8; this report; `reproducibility/independent_run_2/`. ## Relation to the literature, novelty and scope - **Searches (2026-10-14; first stage, the three runs, and at the writing).** arXiv API (k-ranks and maximal k-ranks of binary forms, sums of powers of forms and polynomials, linearly dependent powers of quadratic forms, the Waring number of C[t], Heilbronn's problem, the monic rank, the authors), Crossref, zbMATH, citing works of FLOS, LORS, Carlini–Oneto, BDOV and Newman–Slater in OpenAlex and Semantic Scholar (all 39 titles citing Newman–Slater), nine web searches. - No earlier refutation of Conjecture 1.9, no decomposition of ℓ1ℓ2^{kd−1} with fewer than k terms, and no identity t = Σ_{i