# Verification report — AMR-020-0507 (Bolsinov–Matveev–Miranda–Tabachnikov, Question 5.9: polynomial integrable systems for quadratic Poisson brackets; Bolsinov–Izosimov–Tsonev 2017, Problem 1, for polynomial brackets)

Verification date: 2026-10-17.

**Verdict.** The note gives a **complete negative answer** to the question, with complete proofs.
- **Settled.** The log-canonical bracket {x,y} = xy, {y,z} = √2 yz, {z,x} = √3 zx on R^3 (a quadratic Poisson
  bracket of generic rank 2) has no two Poisson-commuting polynomials f, g with df ∧ dg ≢ 0, so it admits no
  polynomial integrable system in the sense of Section 5 of the source (Theorem A). Two independent proofs: leading
  exponents (main proof, Section 3) and residues on the common level curves (Remark 6.1). The same example shows
  that the answer to Problem 1 of Bolsinov–Izosimov–Tsonev (2017), the same question for polynomial Poisson brackets
  of any degree, can be negative.
- **More.** Theorem C: a log-canonical bracket {x_i,x_j} = λ_ij x_i x_j on R^n admits a polynomial integrable system
  if and only if some maximal isotropic subspace of λ is spanned by vectors of N^n; then monomials form one.
  Theorem B: the explicit criterion on R^3. Theorem D: a log-canonical bracket on R^4 (constants 1, 2^{1/6}, …,
  2^{5/6}) that is symplectic on a dense open set and has no polynomial integrable system. Corollary 5.1: no system
  when the λ_ij (i < j) are linearly independent over Q and n ≥ 3. Proposition 4.2: if the λ_ij are rational
  multiples of one real number, monomials always form a polynomial integrable system, in every dimension (for
  nondegenerate brackets on R^4 two such monomials are given explicitly). Proposition 6.2: counterexamples in every
  degree d ≥ 2.
- **Formulations.** Under the formulation of the introduction of the source (polynomial Casimir functions plus
  commuting integrals) the negative answer is elementary (no polynomial Casimir function for an irrational kernel
  line); the substance is the formulation of Section 5 of the source, which the note answers.
- **Not new mathematics.** The question (Bolsinov, Izosimov, Tsonev 2017, Problem 1, for polynomial brackets;
  Bolsinov, Matveev, Miranda, Tabachnikov 2018, Question 5.9, for quadratic brackets); Sadetov's theorem for linear
  brackets; log-canonical brackets, the bracket of two monomials (Machacek–Ovenhouse, Lemma 3 of arXiv v1) and the
  description of polynomial Casimir functions (polynomial counterpart of Fish–Jordan, Proposition 7.2 of arXiv v3).
- **Not covered.** Quadratic brackets that are not log-canonical are not classified; rational, analytic or smooth
  integrals are not considered (smooth systems exist on the open torus); Problem 5(c) of Bolsinov–Izosimov–Tsonev
  (chaotic Hamiltonians on a polynomial Poisson structure of rank two) is not answered.

The note is unrefereed.

## Statement checked
- **Primary source.** A. Bolsinov, V. S. Matveev, E. Miranda, S. Tabachnikov, "Open problems, questions and
  challenges in finite-dimensional integrable systems", Phil. Trans. R. Soc. A 376 (2018), no. 2131, 20170430,
  doi:10.1098/rsta.2017.0430 (Crossref record checked on 2026-10-17); arXiv:1804.03737v2.
  - Read in the TeX source of v2 (`OpenProblems2018revisedfinal2.tex`, 2865 lines), downloaded from arxiv.org on
    2026-10-17.
  - The question is the `question` environment at lines 1159–1161, in Section 5.2 "Argument shift method and
    Jordan-Kronecker invariants of Lie algebras". All theorem-like environments share one counter per section
    (lines 43–52); counting them in Section 5 gives Theorem 5.1 (l. 1003), Problems 5.2–5.4, Conjecture 5.5,
    Problem 5.6, Question 5.7 (l. 1143, a different question, on algebraic types of pencils), Problem 5.8 (l. 1151,
    "quadratic + linear" pencils), **Question 5.9** (l. 1159).
  - Context (lines 1156–1157): integrable systems "defined on M by a collection of Poisson commuting polynomials";
    for a linear Poisson bracket, equation (5.3) (label `LiePoisson`, line 1094; equations are numbered within
    sections), such a system always exists (Mishchenko–Fomenko conjecture, proved by Sadetov); then the question
    about higher degrees.
  - Definition for degenerate brackets (Section 5, lines 984–986): ½(dim M + corank A) independent commuting first
    integrals, corank at a generic point; equivalently a maximal isotropic span of the differentials. Introduction
    (lines 181–182): commuting functions with differentials linearly independent at almost every point; for a
    Poisson structure of rank 2n on a (2n+r)-manifold, r Casimir functions and n commuting integrals.
  - Nothing in the source restricts the class of quadratic brackets.
- **Corpus record.** ulamai/UnsolvedMath, AMR-020-0507 (dataset version 1.6.0; upstream status open). Its statement
  coincides word for word with the source question; the record labels it "Question 5.7", counting only problems and
  questions. Public texts must cite "Question 5.9 of arXiv:1804.03737v2 (corpus label Question 5.7)".
- **Earlier statement.** A. V. Bolsinov, A. M. Izosimov, D. M. Tsonev, "Finite-dimensional integrable systems: a
  collection of research problems", J. Geom. Phys. 115 (2017) 2–15, doi:10.1016/j.geomphys.2016.11.003; read in full
  by the third verification run in the accepted manuscript of 30 March 2016 (repository of Loughborough University,
  https://hdl.handle.net/2134/23148). Section 2, Problem 1: for a vector space with a polynomial Poisson structure,
  do polynomial integrable systems exist? Same definition as BMMT §5 (formula (1): ½(dim M + corank A) commuting
  functions, independent almost everywhere). Sadetov's theorem is recalled as the positive answer for linear
  structures; the problem is said to remain open for many other polynomial brackets; no counterexample is given.
  BMMT 2018 cites this collection once (l. 1111), for the Generalised Argument Shift Conjecture; its Question 5.9 is
  the quadratic case of Problem 1. The collection says that it is based on an earlier list in Russian (Bolsinov,
  Izosimov, Konyaev, Oshemkov, Trudy Sem. Vektor. Tenzor. Anal. 28 (2012) 119–191), which was not read.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Section 5 definition: ½(n + corank) commuting polynomials with dF_1 ∧ … ∧ dF_k ≢ 0 | no (Theorem A) | Theorem A, Sections 3, 6 |
| Introduction formulation: r polynomial Casimir functions and n commuting polynomial integrals | no, and elementary (no polynomial Casimir function when ker λ ∩ Z^3 = 0); the formulations differ: (1,1,√2) has the system (xz, y) but no polynomial Casimir function | Section 1.3, Example 4.1(c) |
| Complex coefficients for the bracket's constants or the polynomials | no (all proofs work over C) | Theorems A, C |
| Generically symplectic quadratic brackets only (no Casimir functions) | no (Theorem D, R^4) | Theorem D |
| Homogeneous brackets of higher degree d ≥ 2 | no (φ times the bracket of Theorem A) | Proposition 6.2 |
| Linear brackets (d = 1) | yes (Sadetov; not the question) | Section 1.1 |
| Smooth or analytic integrals | not the question; smooth systems exist on the torus | Section 1.2, Remark 6.3 |

## Results in the paper
- **Lemma 2.1** (Jacobi identity, rank 2s on the torus, k = n − s), **Lemma 2.2** ({x^α, x^β} = λ(α,β) x^{α+β};
  polynomial Casimir functions), **Lemma 2.3** (isotropic subspaces: dimension ≤ n − s, maximal ones contain ker λ;
  for n = 3: λ(α,β) = k_λ · (α × β), k_λ = (λ_23, λ_31, λ_12)).
- **Lemma 3.1** (the coefficient of x^{LE p + LE q} in {p,q} is lc(p) lc(q) λ(LE p, LE q), for any total order on
  N^n compatible with addition), **Lemma 3.2** (|LE(W)| = dim W), **Lemma 3.3** (dim span LE(P) ≥ transcendence
  degree; proof by an injective projection to r coordinates), **Proposition 3.4** (structure of Poisson-commutative
  subalgebras).
- **Theorems A, B, C, D**, **Example 4.1** (four brackets on R^3), **Proposition 4.2** (rational constants),
  **Corollary 5.1** (Q-independent constants), **Remark 6.1** (residue proof), **Proposition 6.2** (degree d ≥ 2),
  **Remark 6.3** (what is not covered).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **The program of the note** (`check_note.py`, standard library, about 3 seconds; TOTAL failures: 0): exact
  constants (1, √2, √3, √6 a Q-basis of Q(√2,√3) via the irreducibility of t^4 − 10t^2 + 1; X^6 − 2 Eisenstein;
  Pfaffian θ^5, det λ = θ^10, det A(x) = θ^10 (x1x2x3x4)^2); finite identities (Jacobi identity for symbolic λ,
  n = 3, 4, 5, on coordinate triples and random polynomial triples; Lemma 2.2(a); Lemma 2.3(b); the 3-form identity
  of Remark 6.1; the identities in the proof of Proposition 6.2; the pairs of Example 4.1); the identity of Lemma 3.1
  on random pairs under three monomial orders, with a control order not compatible with addition under which it
  fails; commuting monomial pairs (on R^3, degree ≤ 6: 0 for Theorem A and 191, 0, 86, 0 for Example 4.1(a)–(d);
  on R^4, degree ≤ 4: 0 for Theorem D and 156 for the control); Theorem B on
  all 578 primitive relation vectors in [−4,4]^3 (402 with a system) and the case ρ = 1; Proposition 4.2(b) on all
  13,312 nondegenerate integer matrices with entries in [−2,2]; centralizers certified modulo 10^9+7 (Theorem A
  bracket, degree ≤ 10, five polynomials, and one monomial with degree ≤ 18; Theorem D bracket, degree ≤ 7, two
  polynomials) with three controls.
- **Programs with which the results were first checked** (`original/`): `verify_bracket.py` (4 s) and
  `verify_dim4.py` (28 s), SymPy, exact arithmetic in Q(√2,√3) and Q(θ). Part E of `verify_dim4.py` was replaced:
  its first version tested pairs (f^2 + 3f, f^3 − f) with parallel leading exponents, so it could not fail; the new
  Part E tests the identity of Lemma 3.1 on 30 random pairs (0 failures) and, as a control, under an order not
  compatible with addition (22 failures in 30 pairs, as required).
- **Verification run A** (`verification_run_A/`): symbolic identities for arbitrary functions; centralizers
  certified modulo a prime (Theorem A bracket, degree ≤ 10: 13 polynomials, 5 monomials; Theorem D bracket, degree
  ≤ 7: 6 polynomials, 3 monomials); Theorem B for all 578 primitive relation vectors in [−4,4]^3 and the cases
  ρ = 1, 3; the top-coefficient identity of Lemma 3.1 (2,700 tests, three orders); a search over 1,501 polynomials
  (R^3) and 196 (R^4) for a commuting partner with independent differential: none, while controls produce partners.
- **Verification run B** (`verification_run_B/`): own exact number-field arithmetic; identities; monomial counts;
  Theorem B on 300 random ρ = 2 triples; exact and certified centralizers of about 1,700 polynomials; Lemma 3.1 in
  2,450 cases from brackets with integrable systems under fourteen monomial orders, with failing controls;
  Proposition 4.2(b) on all 108,756 nondegenerate integer matrices with entries in [−3,3].
- **Verification run 2, the third run** (`independent_run_2/`): four programs of its own (symbolic identities
  with arbitrary functions; Lemma 3.1 under up to 31 orders per bracket with failing controls, Lemmas 3.2–3.3;
  Lemma 2.3(a), Proposition 4.2 (a) and (b), Theorem B, the counts; 1,102 centralizers certified modulo
  2146484161), all with TOTAL failures: 0; see the section on this run below.
- **Re-runs.** On 2026-10-17 every program of the package was run again from a copy of `reproducibility/`; the
  results are in `reproducibility/README.md` ("Re-runs"). The third run re-ran all fifteen programs again from an
  extracted copy of `source.zip`: all outputs identical to the recorded ones up to running times
  (`independent_run_2/package_rerun/`).

## Independent verification runs
Two independent verification runs, both AI-assisted, examined the first written version of the results; each wrote
its own programs and did not see the report of the other.

| Item | Run A | Run B |
|---|---|---|
| Statement, definitions, numbering against the source TeX | CONFIRMED (TeX v2 read; line and equation numbers corrected) | CONFIRMED (TeX v2 read) |
| Lemma 2.1 / 2.2 / 2.3 (first version: Lemmas 3.1–3.3, 4.1, 4.5) | CONFIRMED (re-derived; identities for arbitrary functions) | CONFIRMED (re-derived; own exact code) |
| Theorem A, residue proof (Remark 6.1) | CONFIRMED (Steps 1–6 re-derived) | CONFIRMED |
| Theorem A, leading-exponent proof (Lemmas 3.1–3.3, Proposition 3.4, Theorem C) | CONFIRMED (line by line; exactly one product reaches the top monomial) | CONFIRMED (with a control order that fails) |
| Theorem B | CONFIRMED (case analysis; 578 relation vectors) | CONFIRMED (300 random triples) |
| Theorem C | CONFIRMED | CONFIRMED |
| Theorem D, Corollary 5.1 | CONFIRMED | CONFIRMED |
| Proposition 6.2 (first version: Section 5.1) | CONFIRMED (in its first form) | CONFIRMED (in its first form) |
| Proposition 4.2(b) | — | proved and tested by run B (suggestion S2) |
| Programs of the first version | reproduced; Part E of verify_dim4.py vacuous | reproduced; Part E vacuous |
| Novelty | not assessed by run A (one web search: nothing) | no prior answer found; Bolsinov–Izosimov–Tsonev 2017 unread |

No run found a mathematical error.

**Corrections required by run A**, and what was done:
1. Line numbers of the source (context l. 1156–1157, definition l. 984–986) and the equation number (5.3) of the
   Lie–Poisson bracket: used in Section 1.1 of the note and in this report.
2. Hartshorne item numbers: the book could not be checked, so the item numbers were dropped; [Har77] is cited as a
   whole for standard facts in Remark 6.1 only.
3. Part E of `verify_dim4.py` could not fail: replaced by a test that can fail (see "Computations").
4. MSU thesis (d.lib.msu.edu/etd/47927): first described as not read, content unknown; read by the third run when
   a copy became available (see below).
5. Fish–Jordan attribution: located, Proposition 7.2 of arXiv:1611.09721v3 (Section 7), journal version
   J. Pure Appl. Algebra 222 (2018) 2374–2412; cited in Section 1.3 and "Scope and priority".
6. Public texts cite "Question 5.9 of arXiv:1804.03737v2 (corpus label Question 5.7)": abstract, Section 1.1,
   Zenodo description.
7. The Casimir formulation is elementary and the substance is the Section 5 formulation: abstract and Section 1.3.

**Corrections required by run B**, and what was done:
1. Bolsinov–Izosimov–Tsonev 2017 (J. Geom. Phys. 115, 2–15): cited; at first it could not be read (no arXiv
   version in the arXiv author listings of Izosimov and Tsonev; the Crossref full-text links returned an error
   message and metadata only; the publisher's site refused automated access in run B); it was read by the third run
   in the accepted manuscript (see "Statement checked" and the section on the third run). The
   source cites it once, at line 1111, for the Generalised Argument Shift Conjecture. The 2024 problem list
   arXiv:2410.04276 was read in its TeX source (v1): no mention of the question; cited as [BKM26] (published in the
   2024 MATRIX Annals, doi:10.1007/978-3-032-16202-1_8).
2. Sadetov: English translation Doklady Math. 70 (2004), no. 1, 635–638 (Zbl 1378.37101); journal versions of
   arXiv:1206.3882 and arXiv:1611.09721 added.
3. Part E of `verify_dim4.py`: replaced (as A3).
4. Coverage of the search (arXiv export interface and OpenAlex rate-limited; citation check on the Semantic Scholar
   list): stated in "Scope and priority".

**Changes made when the note was written, after runs A and B** (not examined by runs A and B; examined by the
third run, below):
- The leading-exponent proof is the main proof of Theorem A; the residue proof is Remark 6.1, with general
  references.
- Lemma 3.3: the upper bound (d+1)^r is proved by an injective projection of span LE(P) to r coordinates instead
  of a lattice count.
- Proposition 6.2: proved by a direct computation (the trilinear form S of the proof vanishes in dimension three)
  instead of the 1-form criterion; the identities are checked symbolically by `check_note.py` (B8, B9).
- Lemma 2.3(a): every isotropic subspace extends to one of dimension n − s, so "maximal isotropic" in Theorem C
  means "isotropic of dimension n − s".
- Proposition 4.2 (run B's suggestion S2) was added with its proof.
- The program `check_note.py` was written for the note.

## Third independent verification run (verification run 2; the text of the note)
A third independent verification run, AI-assisted, examined the text of the note (12 pages at the start of the run)
and its package on 2026-10-17, as if nothing had been checked before; its programs and outputs are in
`reproducibility/independent_run_2/`.

| Item | Third run |
|---|---|
| Statement, definitions, numbering (BMMT v2 TeX, own download) | CONFIRMED: Question 5.9 (l. 1159–1161), l. 984–986, l. 181–182, l. 1156, equation (5.3); corpus label 5.7 explained |
| Cited statements: MO17 Lemma 3 (arXiv v1), FJ18 Proposition 7.2 (arXiv v3), LLL24 Lemmas 2.2 and 2.12 (arXiv v2), BKM26 (arXiv v1), Sadetov's pages (zbMATH) | CONFIRMED in the TeX sources (theorem counters recounted) |
| Bolsinov–Izosimov–Tsonev 2017, Ovenhouse's thesis (copies became available during the run) | read: the same question in general polynomial form (Problem 1, open there, no counterexample); the thesis contains the monomial bracket and the theorems of Goodearl–Launois and Machacek–Ovenhouse, nothing on commuting families |
| Lemma 2.3(a) with the extension of isotropic subspaces | CONFIRMED (re-derived; 300 random cases) |
| Lemma 3.3, projection proof | CONFIRMED (re-derived; tested on algebras C[f,g]) |
| Proposition 6.2, direct proof | CONFIRMED (re-derived; symbolic check for an arbitrary function φ, Jacobi identity for arbitrary f, g, h; a four-dimensional control fails) |
| Proposition 4.2 | CONFIRMED; the sentence calling the rational case undecided in dimension ≥ 4 replaced by a proof that monomials always form a system when the λ_ij are rational multiples of one number (supplied by this run; tested on 400 random rational matrices, n ≤ 9); part (b) tested on all 13,312 and 108,756 nondegenerate integer matrices (entries in [−2,2], [−3,3]) |
| All other lemmas, Theorems A–D, Example 4.1, Corollary 5.1, Remark 6.1 | CONFIRMED (re-derived line by line) |
| Lemma 3.1 | CONFIRMED; 2,140 tests under up to 31 orders compatible with addition per bracket (including the anti-graded order and weight orders with weights of both signs); three non-compatible control orders fail |
| Theorem B | CONFIRMED; 578 primitive ν, 402 with a system, agreement with two independent searches |
| Centralizers | 1,102 certified modulo 2146484161 (brackets of Theorems A, D and Example 4.1(b), (d)); controls give larger kernels |
| Package | all 15 programs re-run from the archive: outputs identical to the recorded ones up to running times |
| Novelty | no earlier answer found (arXiv export interface and OpenAlex listings refused again; zbMATH, Crossref, one web search); the question is older than BMMT 2018 |

It found no error in the mathematics. **Corrections required by the third run**, and what was done:
1. Credit Problem 1 of Bolsinov–Izosimov–Tsonev (2017) as the earlier statement of the question for polynomial
   brackets, with BMMT Question 5.9 as its quadratic case: abstract, Sections 1.1 ("An earlier formulation") and
   1.3, "Scope and priority", bibliography note of [BIT17].
2. Cite the 2012 Russian list on which that collection is based, as not read: [BIKO12].
3. Report the content of Ovenhouse's thesis [Ove19] (Lemma 1 credited next to MO17 Lemma 3); add to Remark 6.3 that
   the note does not answer Problem 5(c) of [BIT17].
4. Section 7: the counts of Part (D) are 0 for Theorem A and 191, 0, 86, 0 for Example 4.1(a)–(d) (the text had
   assigned 86 to (b)); Part (G) has five polynomials with D = 10 and one monomial with D = 18. Corrected in the note,
   in this report and in the package README.
5. Proposition 4.2: part (a) now states and proves that for λ a real multiple of a rational matrix, monomials with
   positive exponents form a polynomial integrable system in every dimension; part (b) keeps the explicit pair for
   n = 4; the sentence "Whether there are counterexamples with rational constants in dimension four and rank two,
   or in dimension at least five, is not decided here" was removed; Section 1.2 adjusted.
6. "Scope and priority": the question was asked first in 2017 in the general polynomial form, open there, with no
   counterexample; what was read and the searches updated (ten web searches in all).
7. Section 1.3: the novelty sentence now separates the negative answer (Theorem A) from the other results, and
   calls Propositions 4.2 and 6.2 simple complements.
8. This run recorded in the Verification paragraph, in Section 7 and here; `independent_run_2/` added.
9. Zenodo metadata: publication date 2026-10-17; description = the final abstract.

## Relation to the literature, novelty and scope
- **Read.** The source in the TeX of v2 (introduction, Section 5, bibliography); Bolsinov–Konyaev–Matveev,
  arXiv:2410.04276v1 (TeX searched: no quadratic Poisson brackets, no polynomial integrable systems, no Sadetov);
  Machacek–Ovenhouse, arXiv:1703.06521v1 (Sections 1–4: no rational change of coordinates makes a log-canonical
  bracket constant or linear; Lemma 3 is the bracket of two Laurent monomials); Fish–Jordan, arXiv:1611.09721v3
  (Section 7: Definition 7.1 with the formula for {x_i, x^m}; Proposition 7.2, trivial Poisson centre of the
  Laurent polynomial algebra iff no nonzero integer kernel vector, derived there from a lemma of Vancliff);
  Li–Li–Lu, arXiv:2411.18904v2 (abstract, Sections 1–2 on integrable systems and lowest-degree terms; Lemma 2.2 is
  the bound on the size of an independent involutive family; Lemma 2.12 passes from log-canonical pairs to their
  lowest-degree terms; no quadratic brackets); Bolsinov–Izosimov–Tsonev 2017 in full, in the accepted manuscript
  (Problem 1 is the same question for polynomial brackets; open there; no counterexample; the other problems do not
  contain an answer); N. Ovenhouse, PhD thesis, Michigan State University, 2019 (Chapter 1: log-canonical brackets,
  the bracket of two Laurent monomials, the theorem of Goodearl–Launois that no rational f, g satisfy {f,g} = 1,
  the main theorem of Machacek–Ovenhouse; Chapter 10; the other chapters, on pentagram maps and non-commutative
  integrability, searched: no commuting families of polynomials for log-canonical brackets). Bibliographic data
  checked with Crossref (BMMT, BIT, MO, FJ, Bolsinov 2016, Hartshorne, BKM26).
- **Not read.** Sadetov (2004) (cited as in the source, pages from zbMATH); the 2012 Russian list of Bolsinov,
  Izosimov, Konyaev and Oshemkov (cited as in Bolsinov–Izosimov–Tsonev 2017; the question may already be there).
- **Searches** (2026-10-17, for the first written version, by the three runs and when the note was written): arXiv
  search pages and export interface (many export requests refused: HTTP 429/503), the arXiv author listings of
  Izosimov and Tsonev, the 62 works citing the source in Semantic Scholar (titles), zbMATH, Crossref, ten web
  searches. OpenAlex listings were refused (rate limit). No printed answer to the question was found in the sources
  searched; a search that finds nothing is not a proof of novelty. The question was asked first in 2017 in the
  general polynomial form (Bolsinov–Izosimov–Tsonev, Problem 1), open there. The counterexample is elementary, and
  experts may regard it as known. No priority is claimed.
- **Scope.** The note answers the question negatively (and shows that the answer to Problem 1 of
  Bolsinov–Izosimov–Tsonev can be negative) and characterises the log-canonical brackets with a polynomial
  integrable system; other quadratic brackets are not classified.

## 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/