# Verification report — AMR-046-0027 (Gasull, "A moments problem II": the numbers N(k))

Verification date: 2026-09-30. The paper was revised on the same day after a second independent verification run,
and two sentences were made more precise after a final readiness check (see the end of the section on the
verification runs).

**Verdict.** A scoped partial answer. Part (i) of Gasull's question is answered affirmatively: N(k) exists for every
k. The proof uses Hilbert's basis theorem and the known fact (Pakovich; Françoise–Pakovich–Yomdin–Zhao) that the
moments ∫_0^1 f^n dx, n ≥ 1, of a nonzero polynomial cannot all vanish, and it is not effective. Part (ii) is
answered for k ≤ 3: N(1) = 1, N(2) = 2 and N(3) = 3; the value N(3) = 3 rests on an exact computer-assisted
certificate. In general N(k) ≥ k. For k ≥ 4 the value of N(k), and any explicit upper bound, remain open. The note is
unrefereed.

## Statement checked
- **Primary source.** A. Gasull, "Some open problems in low dimensional dynamical systems", arXiv:2012.02524 (v1, the
  only version); published in SeMA Journal 78 (2021) 233–269, doi:10.1007/s40324-021-00244-3.
  - The TeX source was fetched anonymously from arXiv and read. The problem is Problem 27 of the preprint, in the
    section "Problems involving polynomials", subsection "A moments problem". It begins "Is there a value N(k) such
    that if the following finite set" of moment conditions holds.
  - Setting: f ∈ C[x] a polynomial with k monomials, M_n = ∫_0^1 f(x)^n dx for 1 ≤ n ≤ N(k), conclusion f = 0.
    Part (ii) asks for N(k) or a good upper bound. The exponents are therefore nonnegative integers and the
    coefficients complex.
  - The preceding Problem 26 (van den Essen) asks whether all moments m ≥ 1 vanishing forces f = 0, in several
    variables. For one variable Gasull cites Pakovich, Mosc. Math. J. 13 (2013) 693–731.
- **Corpus record.** ulamai/UnsolvedMath, AMR-046-0027 (status `open`). Its statement agrees with the source.
- **Pakovich's theorem.** Read in arXiv:0910.2105v1 (the only arXiv version): Theorem 3.4 and Corollary 3.5. The
  published version is behind a paywall and was not consulted, so its numbering was not checked; the paper says so.
- **Credit for Corollary 3.2** (checked for the revision).
  - A. van den Essen and W. Zhao, "Mathieu subspaces of univariate polynomial algebras", arXiv:1012.2017v1 (the only
    version); J. Pure Appl. Algebra 217 (2013), no. 7, 1316–1324, doi:10.1016/j.jpaa.2012.10.006 (Crossref confirms
    the volume, issue, pages and DOI). The TeX source and the PDF were fetched anonymously from arXiv and read.
    - The theorem on the Jacobi weights (1−t)^α(1+t)^β on (−1, 1) is printed as **Theorem 6.8** in the arXiv PDF.
      Its label in the TeX source is `T6.7`, and the second verification run cited it as Theorem 6.7; the printed
      Theorem 6.7 is a different result.
    - Part (i): if α, β ∈ N (N contains 0 there), the radical of V_B(σ) is 0, that is, ∫ f^m w dt = 0 for all
      large m forces f = 0. The paper says that this follows from Theorem 3.4 and Corollary 3.5 of Pakovich's preprint.
    - Part (ii): the same for α = β = λ − 1/2 with λ ∈ ½N; this is attributed to Proposition 4.2 of FPYZ.
    - The proof of Theorem 6.2 there mentions an unpublished proof by M. Boyarchenko for B = [0, 1] with the Lebesgue
      measure ("but see [FPYZ]").
    - The journal version was not consulted; the paper uses the arXiv numbering and says so.
  - J.-P. Françoise, F. Pakovich, Y. Yomdin and W. Zhao, "Moment vanishing problem and positivity: some examples",
    Bull. Sci. Math. 135 (2011), no. 1, 10–32, doi:10.1016/j.bulsci.2010.06.002 (Crossref confirms).
    - Access: the publisher's page returned a bot check (HTTP 403) and was not used, and the copy on Pakovich's web
      page did not respond. The published version, with the journal's pagination, was read in an archived copy of
      that web-page file in the Internet Archive.
    - Section 4, "Boyarchenko's proof and some other results": Theorem 4.1 is built on Boyarchenko's argument.
      Corollary 4.1 states that for a ≠ b in C and f ∈ C[z], ∫_a^b f^m dz = 0 for all m ≥ N forces f = 0; the
      authors note that it also follows from Theorem 3.4 of Pakovich's preprint. Proposition 4.2 treats the weights
      (1 − z²)^{λ−1/2} with λ ∈ ½N and is proved with Pakovich's Theorem 3.4. The acknowledgements thank Boyarchenko
      for a sketch of his proof of Corollary 4.1.
  - Conclusion: after z = (1+t)/2, Corollary 3.2 of the paper (the weight s z^{s−1} on [0, 1]) is the case α = 0,
    β = s − 1 of vdEZ13, Theorem 6.8(i). The Lebesgue case s = 1 is also FPYZ11, Corollary 4.1, and it follows from
    FPYZ11, Proposition 4.2 with λ = 1/2 (vdEZ13, Theorem 6.8(ii)). The paper now says this in the abstract, in the
    introduction, in "Related work" and in a paragraph after the proof of Corollary 3.2.

## Readings
| Reading | Result | Where |
|---|---|---|
| "k monomials" = at most k nonzero terms (N(k)) | N(k) < ∞ for all k (not effective); N(k) ≥ k; N(1) = 1, N(2) = 2, N(3) = 3 | Thm 1.3 |
| exactly k nonzero terms (N^=(k), with N(k) = max_{j≤k} N^=(j)) | N^=(k) < ∞; N^=(k) ≥ k for k ≤ 4; N^=(1), N^=(2), N^=(3) = 1, 2, 3 | Thm 1.3 |
| nonnegative rational exponents | the same bound N_0(k) works | Thm 3.3 |
| real exponents, k = 3, −1/3 < e_1 < e_2 < e_3 | M_1 = M_2 = M_3 = 0 forces f = 0 | Thm 1.4 |
| real exponents, k = 4, e_1 > −1/4 | numerically false for some e_1 < 0 (not proved) | Sec. 7 |

## Results in the paper
- **Theorem 3.3** (part (i)). For each k there is N_0(k) such that for distinct nonnegative rational exponents,
  M_1 = ... = M_{N_0(k)} = 0 forces c = 0.
  - The exponents are treated as indeterminates. P_n = D_n M_n (denominators cleared) lies in
    Q[e_1..e_k, c_1..c_k], and the ideal chain (P_1, ..., P_N) is stationary from N_0(k) on (Hilbert's basis
    theorem). Hence the vanishing of the first N_0(k) moments gives the vanishing of all moments, since D_n(e) ≥ 1.
  - Corollary 3.2 then gives f = 0. It applies Pakovich's Corollary 3.5 (arXiv:0910.2105v1) with P(z) = f(z^s),
    q(z) = s z^{s−1} and γ = [0,1]: the moments for i ≥ 1 vanishing would force ∫_0^1 q dz = 1 to vanish.
    Corollary 3.2 is a special case of vdEZ13, Theorem 6.8(i), and for s = 1 it is FPYZ11, Corollary 4.1 (see
    above); the paper says so.
  - Remark 3.4: N_0(k) is not effective. Remark 3.5: for bounded degree, Batenkov–Binyamini (JDE 259 (2015),
    Theorem 1) with Pakovich gives M_1 = ... = M_{3d²−5d+5} = 0 ⇒ f = 0 for deg f ≤ d. This bound is not uniform in
    the degree.
- **Proposition 4.1.** N(k) ≥ k for every k (k − 1 forms in k variables have a common nontrivial zero). The zero has
  all coordinates nonzero when N(k−1) ≤ k−1, so N^=(k) ≥ k for k ≤ 4. N(1) = 1 and N(2) = 2.
- **Theorem 1.4 / Proposition 5.1** (k = 3, exact computation).
  - With c_i = (1+e_i)y_i and (y_1, y_2, y_3) = (1, r, −1−r), the cleared moments P_2(r), P_3(r) have coefficients in
    Z[e_1, e_2, e_3]. The leading coefficient of P_2 is D_2 ∫ g² > 0.
  - The Sylvester resultant R = Res_r(P_2, P_3) (5 × 5 determinant, formal degrees 2 and 3) has 10426 terms, total
    degrees 18–45, and **content 4**. The constants below refer to this normalisation.
  - (i) R(u, u+v, u+v+w) = 4 (vw(v+w))^6 R_1, where R_1 has 3290 positive integer coefficients and R_1(0) = 29.
  - (ii) R(−1/3+u, ...) = 4 (vw(v+w))^6 R_2, where R_2 has 3115 positive rational coefficients and its lowest pure-u
    term is (200000/177147) u^6. For the computation the exponents are written as E = 3e: each factor 1 + β·e becomes
    (3 + β·E)/3 and each 1 + e_i becomes (3 + E_i)/3; clearing the 3's multiplies P_2 by 3^7, P_3 by 3^12 and R by
    3^(3·7+2·12) = 3^45.
  - (iii) R(u, u+1+v, u+2+v+w) has 13400 positive integer coefficients, with constant term
    R(0,1,2) = 82556485632000 = 4 · 20639121408000. (Divided by the content, this is 20639121408000.)
  - Hence R > 0 for all real −1/3 < e_1 < e_2 < e_3, so P_3 does not vanish at a root of P_2. The case y_1 = 0 is
    excluded because c is then a multiple of a real vector, where M_2 ≠ 0.
- **Lemma 6.1 / Corollary 6.2.** If (e_1..e_{k−1}) with e_1 ≥ 0 has property (H_{k−1}), then (e_1..e_{k−1}, E) has
  (H_k) for all large E. So for every k, sufficiently lacunary exponents satisfy (H_k). The thresholds are not
  explicit.
- **Section 7**, finite-range computations and numerics.
  - Exact certificates: k = 4 for all 249,900 integer sets with e_4 ≤ 50 and 2000 random sets with e_4 ≤ 10^4;
    k = 5 for all 6188 sets with e_5 ≤ 16 and 300 random sets with e_5 ≤ 1000. These are labelled as finite-range
    computations. For k = 5 the paper now describes all four charts, and it justifies the elimination in F_p by the
    p-integrality check (of the coefficients of the forms and of the numbers d_j; 1/d_1 is p-integral because
    d_1 = M_2(e; (1+e_1, 0, 0, 0, −(1+e_5))) has a numerator smaller than p for integer exponents with
    e_5 ≤ 1000), the degree bounds
    36, 42 and 64 (from the total degrees 6, 6, 7, 8, 8) and interpolation at distinct nodes.
  - Numerical margins, and a numerical failure of (H_4) for real exponents with e_1 < 0, labelled as numerical:
    - ρ_4 ≈ 2.0·10^−14, at the floor of double precision, at
      e* = (−0.18271900153639156, −0.1809796744363755, 769.0851161513428, 825.7277931734245), where 1 + 4e_1 ≈ 0.269.
      The rounded point (−0.18272, −0.18098, 769.085, 825.728) gives 9.4·10^−6, so the paper now prints all digits.
    - With e_1, e_2 fixed, Newton's method in (e_3, e_4) converges quadratically to
      (769.08511614717282823869…, 825.72779317804078768512…). There ρ_4 is at the level of the working precision: below
      3·10^−44 in 60-digit and below 10^−64 in 80-digit arithmetic.
    - The winding number of M_4 along the branch is 1 on circles of radius 0.5 and 2 about (e_3*, e_4*), and on the
      circle of radius 10^−6 about that point.

## Computations (exact unless stated; scripts and outputs in reproducibility/)
- **Lead** (`lead/`).
  - `check_theorem_C.py` is standard library only, written independently of the other code, and runs in about one
    minute.
    It checks every statement of Proposition 5.1:
    - it recomputes P_2, P_3 and R from the definitions;
    - it checks the 13-term generic Sylvester formula symbolically, via A²R = λ²C − λμB + Aμ², where
      A² P_3 ≡ λ r + μ mod P_2; this also proves R = A³ P_3(ρ_1) P_3(ρ_2);
    - it checks the counts, the degrees and the content 4;
    - it checks identities (i)–(iii) by exact division and multiplication back, with all coefficients positive;
    - it checks the constants 29, 200000/177147 and 82556485632000;
    - it checks R = D_2³ D_3² Res(M_2, M_3) at 60 random rational points, with M_n computed from the integral
      formula;
    - it checks gcd(M_2(r), M_3(r)) = 1 for all 2300 integer triples with e_3 ≤ 24.
    - Its certificate file `R1_uvw.txt` has exactly the same 3290 coefficients as the finder's `k3_Rprime_uvw.txt`
      (the header and the order of the lines differ).
  - `check_k4_range.py` is a separate implementation of the k = 4 certificate, using Sylvester resultants in u_1 and
    rational interpolation instead of reduction. It certifies all 249,900 sets with e_4 ≤ 50.
  - `check_k4_real_zero.py` (added for the revision; standard library only, 80-digit decimal arithmetic, about 20 s)
    reproduces the real k = 4 example of Section 7 with a different elimination: the norm Res_{y_2}(M_2, M_3) is
    computed from exact Sylvester determinants at 9 points and exact interpolation, and its roots by the Durand–Kerner
    method. It gives ρ_4 = 9.392·10^−6 at the rounded point and 2.034·10^−14 at e*. Its Newton iteration in (e_3, e_4)
    reaches ρ_4 = 9.6·10^−65 at a point that agrees with the 60-digit value of the second verification run in all 39
    digits printed. At that point the real Jacobian of (Re M_4, Im M_4) with respect to (e_3, e_4) has determinant
    1.2·10^−24 > 0, and the winding number on the circle of radius 10^−6 is 1 (adaptive steps). Numerical, not a
    proof.
  - `reruns/`: the finder's k = 4 and k = 5 programs, re-run on 2026-09-30, reproduce every entry of Table 1 and
    the negative controls. The numerical k = 4 example with e_1 < 0 is also reproduced (ρ_4 ≈ 2·10^−14 at the full
    double-precision point e*).
- **Finder** (`claimant/`).
  - First computation of R and of the certificates (i)–(iii).
  - R agrees with exact Sylvester determinants at 5 + 300 random triples.
  - M_1 = M_2 = M_3 = 0 was solved exactly in Q(√D) for all integer triples with e_3 ≤ 60, and no solution was
    found.
  - k = 4 and k = 5 finite-range certificates, with negative controls.
  - Numerical margins and the numerical failure for e_1 < 0.
  - Re-running `scripts/k3/` in the release layout reproduces the three certificate text files byte for byte.
- **First independent verification run** (`verifier/`, AI-assisted, separate code).
  - R was recomputed by memoized Laplace expansion in the coordinates of (iii), (i) and (ii). This gives 13400
    nonnegative coefficients with constant 82556485632000 and content 4; divisibility by (vw(v+w))^6 with cofactor
    of 3290 nonnegative coefficients and constant 116 = 4·29; and 3115 nonnegative coefficients with lowest pure-u
    term of degree 6.
  - R was validated against direct rational Sylvester determinants at 80 points.
  - A Newton-refined numerical check gives ρ_4(1,2,3,4) ≈ 0.0984. In its refined list for (1,2,3,4) one root was
    found twice; the six values are the pairs 0.09837, 0.5172 and 0.61716 (noted in the README).
- **Second independent verification run** (`independent_run_2/`, AI-assisted, separate code; see its README).
  - R and the identities (i)–(iii) recomputed from equations (3) and (4), with all counts and constants. Its R and
    R_1 agree coefficient by coefficient with the shipped `k3_R_abc.txt` and `R1_uvw.txt`. Its R_2 is one quarter of
    the shipped `k3_Rprime_shifted_uvw.txt`, which, as its header says, is not divided by the content 4.
  - A direct exact test of (H_3), without R, on 17,660 triples (all integer triples with e_3 ≤ 40, random integer
    triples with e_3 ≤ 10^6, rational triples with e_1 > −1/3 near −1/3, near confluence and with very large gaps):
    no failure.
  - A different certificate for Table 1: the Macaulay matrix of M_2, …, M_k in degree k(k−1)/2 + 1 has full rank
    modulo 2^31 − 1, which is equivalent to (H_k). It certifies all 249,900 k = 4 sets with e_4 ≤ 50 and all 6188
    k = 5 sets with e_5 ≤ 16, as well as random sets with large exponents. Negative controls fail as they must.
  - 60-digit numerics for Section 7 (the values above), and a spot scan of ρ_4 over 1106 integer sets (minimum
    0.09837 at (1,2,3,4)).
  - Reruns of the previous release package: all shipped outputs were reproduced, and the three Zenodo hashes matched
    the metadata.
  - On 2026-09-30 all of its scripts were rerun from the copies in `independent_run_2/`, and every recorded output
    was reproduced up to timings.

## Independent verification runs and required fixes

### First independent verification run (AI-assisted)

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (source TeX re-read; integer exponents, complex coefficients, moments M_1..M_{N(k)}) |
| Theorem 3.3 (with Pakovich) | CONFIRMED (Pakovich, arXiv:0910.2105v1, Thm 3.4 and Cor 3.5 read) |
| Proposition 4.1, N(1), N(2), Lemma 6.1 | CONFIRMED (the AI-assisted verification run re-derived the proofs step by step) |
| Theorem 1.4 / N(3) = 3 | CONFIRMED (own exact code) |
| k = 4, 5 certificates | not re-run in that verification run; re-run by the lead and, for k = 4, re-implemented; certified again by the second verification run (below) |
| Classification | partial result; part (ii) open for k ≥ 4 |

All four required fixes of the first run were applied:
1. Theorem 3.3 is stated as non-effective (theorem statement, Remark 3.4, abstract, "Scope and priority"). Part
   (ii) is stated to be open for k ≥ 4.
2. Pakovich is cited precisely: Corollary 3.5, deduced from Theorem 3.4, of arXiv:0910.2105v1. The published version
   could not be consulted, and the paper says that its numbering may differ.
3. The k = 4 (e_4 ≤ 50) and k = 5 (e_5 ≤ 16) results are labelled as finite-range computations (abstract, Section 7,
   Table 1, "Scope and priority"). The k = 4 real-exponent failure is labelled as numerical.
4. The normalisation of the resultant is stated explicitly: the Sylvester determinant (4) of the cleared moments (3),
   with content 4. The constant terms are given in that normalisation, together with the primitive value
   20639121408000.

### Second independent verification run (AI-assisted)

No mathematical error was found. The run recommended a minor revision: wording, credit, reproducibility of one
numerical example, and two justifications.

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (arXiv source re-read; Problem 27 in the preprint numbering; the corpus record agrees) |
| Proofs (Lemma 2.2, Corollary 3.2, Theorem 3.3, Remarks 3.4 and 3.5, Proposition 4.1, Section 5, Lemma 6.1, Corollary 6.2, the certificate logic of Section 7) | CORRECT (re-derived step by step) |
| Proposition 5.1 / Theorem 1.4 | REPRODUCED with new code; (H_3) also tested directly on 17,660 triples, no failure |
| Table 1 | all sets with e_4 ≤ 50 (k = 4) and e_5 ≤ 16 (k = 5) certified by the Macaulay-matrix criterion |
| Section 7 numerics | CONFIRMED in 60-digit arithmetic; the rounded point printed in the paper did not reproduce the quoted ρ_4 |
| Release package | reruns reproduce the shipped outputs; hashes matched |
| Novelty | no prior statement of Theorem 1.3(a) or of N(3) found; credit for Corollary 3.2 incomplete |

Required fixes of the second run, and how they were applied:
1. **Wording.** In the Verification paragraph, item 3 was reworded. It now says that the AI-assisted verification
   run re-derived step by step the proofs of Theorem 3.3 (including the use of Theorem 3.1), Proposition 4.1 and
   Lemma 6.1. The same change was made in the table above. The second run is described as a
   second AI-assisted verification run (Verification, item 5). The web-search count in "Scope and priority" is now
   three (finder, author, second verification run), in agreement with the request logs.
2. **Real k = 4 example (Section 7).** The paper now gives all digits of e* and the value 9.4·10^−6 at the rounded
   point. It says that 10^−14 to 10^−13 is the double-precision floor, gives the high-precision zero and the winding
   number on the circle of radius 10^−6, and keeps the label "numerical, not proved". These numbers were reproduced
   by `independent_run_2/r2_k4_negative_hp.py` (60 digits) and by `lead/check_k4_real_zero.py` (80 digits,
   different elimination). One correction to the requested wording: the 60-digit values of ρ_4 at the converged
   point range from 1.5·10^−45 to 2.0·10^−44, so the paper says "below 3·10^−44" rather than "below 10^−44".
3. **Credit.** Corollary 3.2 is now credited as described under "Credit for Corollary 3.2". The theorem of van den
   Essen and Zhao is cited as Theorem 6.8 (arXiv numbering), not Theorem 6.7.
4. **Proposition 5.1(ii).** The scaling sentence now says what is done: E = 3e, the factor 3^7 for P_2, 3^12 for P_3,
   and hence 3^45 for R. These exponents agree with `lead/check_theorem_C.py`, where `moment_polys` returns
   3^(n+N_n−1) P_n with N_2 = 6 and N_3 = 10.
5. **k = 5 (Section 7).** The justification of the F_p test and the description of charts B, C and D follow
   `claimant/scripts/k5_exact_cert.py`. That program checks p-integrality and stops the certificate otherwise, asserts
   the total degrees 6, 6, 7, 8, 8, and interpolates at 37, 43 and 65 distinct nodes.

Optional suggestions: the following were applied.
- Issue numbers were added to the bibliography (PM09 no. 3, FPYZ11 no. 1, vdEZ13 no. 7).
- The duplicated root in the `verifier/k4num2.py` output is explained in the README.
- The README now says "the same coefficients" for `R1_uvw.txt`.
- The Macaulay criterion is mentioned in the Verification paragraph.

The other suggestions were not applied in this revision: the convention N^=(k) = ∞, the phrase "non-explicit
thresholds" in the abstract, further background citations, and an interval-Newton proof of the k = 4 real failure.

### Final readiness check (AI-assisted)

A final AI-assisted check of the revised package found no mathematical error and led to two small corrections of
wording. It re-checked the following.
- The required fixes of both verification runs are applied.
- In arXiv:1012.2017v1 (PDF and TeX source fetched anonymously) the theorem on the Jacobi weights is printed as
  Theorem 6.8; the paper cites it by that number and says in the bibliography that it uses the arXiv numbering.
- Corollary 4.1, Theorem 4.1 and Proposition 4.2 of FPYZ11 were read again in the archived copy of the published
  version, and the three DOIs of JPAA 217(7), Bull. Sci. Math. 135(1) and PLMS 99(3) were confirmed on Crossref.
- A fresh extraction of `source.zip` builds, and `lead/check_theorem_C.py` and `lead/check_k4_real_zero.py`
  reproduce their recorded outputs up to timings.
- A short separate exact check was run; its script is not part of this package. It computes R from equations (3)
  and (4) at rational points. There R agrees with the shipped certificate polynomials of Proposition 5.1(i) and
  (ii) at 12 random rational points each. It checks R(0,1,2) = 82556485632000 and the constant 200000/177147. It
  finds R(e) > 0 on 288 random triples with e_1 > −1/3 (integer, rational, near −1/3, near confluence and with very
  large gaps), and R = D_2^3 D_3^2 Res(M_2, M_3) on 60 of them. It also recounts the exponent 45 = 3·7 + 2·12.

Corrections:
1. **Section 7, k = 5.** The paper said that every rational coefficient that occurs is checked to be p-integral.
   The program checks the coefficients of the forms and the numbers d_j. The coefficients −d_j/d_1 are p-integral
   for another reason: 1/d_1 is p-integral. In the program's coordinates d_1 = M_2(e; (1+e_1, 0, 0, 0, −(1+e_5))).
   Over all integer pairs 0 ≤ e_1 < e_5 ≤ 1000 the largest numerator of d_1 is 296887552592, which is smaller than
   p = 2^61 − 1. The paper now says this. The certificates themselves are unchanged.
2. **Section 7, the real k = 4 example.** The high-precision zero was printed with rounded digits followed by "…".
   It is now printed with truncated digits: (769.08511614717282823869…, 825.72779317804078768512…). These digits
   agree with the 60-digit and the 80-digit computations.

Also, in `independent_run_2/README.md` the description of the scaling in (ii) now says that the denominators 3 are
cleared, as in the paper. No web search was made in the final check.

## Relation to the literature, novelty and scope
- **Searches (September 2026).**
  - arXiv API: vanishing moments with monomials, fewnomials, lacunary or sparse polynomials; the polynomial moment
    problem; the Bautin index; the image conjecture and Mathieu subspaces; the Gaussian moments conjecture. The
    second verification run repeated these searches and added the moment Bautin index and Gasull's recent papers.
  - Crossref, OpenAlex, zbMATH (including zbMATH's list of works citing Gasull's survey, Zbl 1487.37024).
  - OpenAlex lists 39 works citing Gasull's survey and 14 entries (13 distinct DOIs) citing Pakovich (2013). None
    treats this problem; one treats Gasull's Problems 28 and 29.
  - Three web searches were made: one by the finder, one for this note and one in the second verification run. An
    earlier start of the second run, which was interrupted, also attempted one; it returned no results because of a
    session limit. No web search was made for the revision or for the final check.
  - Nothing states part (i) or gives a value of N(k) for k ≥ 3.
- **Related work credited.**
  - Pakovich (2013), whose theorem answers a question of Zhao (2010), and Françoise–Pakovich–Yomdin–Zhao (2011,
    Section 4, including Boyarchenko's argument), for the case of all moments.
  - van den Essen–Zhao (2013, Theorem 6.8), of which Corollary 3.2 is a special case.
  - Pakovich–Muzychuk (2009), the solution of the polynomial moment problem.
  - Batenkov–Binyamini (2015), the moment Bautin index: an explicit bound for bounded degree, and finiteness by
    Noetherianity.
  - Cima–Gasull–Mañosas (2013), cited by Gasull next to the problem.
  - van den Essen (2000), Derksen–van den Essen–Zhao (2017) and Françoise (2014), the connection with the Jacobian
    conjecture.
- **Novelty.** The Noetherian argument for part (i) is routine in spirit, and the paper says so. Theorem 1.4, Lemma
  6.1 and the computations appear to be new ("we found no earlier ..."). This negative search is not a proof of
  priority.
- **Scope.** Proved:
  - part (i), yes, not effective;
  - N(k) ≥ k;
  - N(1) = 1, N(2) = 2, N(3) = 3;
  - Theorem 1.4 for real exponents greater than −1/3;
  - Lemma 6.1 and Corollary 6.2.

  Computed on finite ranges: k = 4, 5. Numerical: the margins and the k = 4 failure for e_1 < 0. Open: N(k) for
  k ≥ 4, and any explicit upper bound.

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