# Verification report — AMR-014-0019 and AMR-021-0007 (isolated real zeros of sums of squares)

Verification date: 2026-09-30.

**Verdict.** Scoped partial answer to both conjectures. Proved: every real linear system of forms of degree k on P^3
has at most k^3 isolated real base points (every k), and every real linear system of quadrics on P^4 has at most 16.
Consequently B'_{2k,4} = k^3 for every k (the whole row m = 4 of the Froberg-Lundqvist-Oneto-Shapiro conjecture, with
n = degree and m = number of variables), B'_{4,5} = 16, and the Ottaviani-Shapiro count satisfies ~#(2k,3) = k^3 for
every k and ~#(4,4) = 16. The paper also gives a short proof of B'_{4,4} = 8. Both conjectures remain open in general,
for instance for B'_{6,5}, B'_{4,6}, ~#(6,4) and ~#(4,5). Two independent AI-assisted verification runs found no
mathematical error or gap; the second one checked the preceding version line by line and wrote its own code, and the
present version differs from it only by the clarifications and small corrections it asked for or suggested. The note
is unrefereed.

## Statement checked
- **AMR-014-0019.** R. Froberg, 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), subsection
  "Non-negative forms". The TeX source was read.
  - The conjecture reads "For any given pair (n,m) with even n, B'_{n,m}=(n/2)^{m-1}".
  - B'_{n,m} is the supremum of |Z(p)| over sums of squares p of degree n with finitely many real zeros.
  - The sentence that defines the cones speaks of forms of degree m in n variables, but the formula, the phrase "with
    even n", the list of Hilbert's cases and the remark that the case m = 3 is proved fix n = degree, m = number of
    variables. The paper uses this convention throughout and says so in Section 1. (Re-checked in the TeX source of
    arXiv:1801.01692, fetched anonymously during this revision.)
  - A question about B'_{4,4} occurs only in a commented-out line of the TeX source. It is not part of the published text,
    and the paper does not refer to it.
- **AMR-021-0007.** B. Shapiro, "Problems around polynomials: the good, the bad and the ugly...", Arnold Math. J. 1 (2015)
  91-99, doi:10.1007/s40598-015-0008-4 (arXiv:1503.05295), part III. The TeX source was read.
  - ~#(2k,l) is the maximal number of isolated zeros of a nonnegative polynomial of degree 2k in l variables that is a
    sum of squares of polynomials of degree at most k.
  - The conjecture, attributed to Ottaviani and Shapiro, is ~#(2k,l) = k^l for any number of variables.
  - Shapiro credits l = 2 to Choi-Lam-Reznick and records the elementary bounds k^l <= ~#(2k,l) <= (2k-1)^l.
- **Choi-Lam-Reznick, Math. Z. 171 (1980) 1-26, doi:10.1007/BF01215051.** Read in full, in the GDZ scan of the journal
  version. Relevant content:
  - Prop. 4.1: B'_{n,m} >= (m/2)^{n-1} (their convention; the grid example).
  - Thm 4.6: B'_{3,m} = m^2/4, i.e. B'_{2k,3} = k^2 in our convention.
  - Prop. 4.13: 8 <= B'_{4,4} <= 10 <= B_{4,4} <= 11.
  - End of Section 4 (p. 13): B'_{4,4} = 8 seems provable by Bezout arguments for quadric surfaces; no details given.
  - No other upper bound for B' with at least four variables.
- **Dressler, J. Pure Appl. Algebra 225 (2021) 106602, doi:10.1016/j.jpaa.2020.106602 (arXiv:1909.06707).** TeX read.
  The remark that closes Section 4 states B'_{4,4} = 8; the corresponding item of her list of known values is commented
  out; she writes that B and B' are open for general n and d. Her indices put the number of variables first (her
  B'_{5,4} is our B'_{4,5}).
- **Blekherman-Hauenstein-Ottem-Ranestad-Sturmfels, Compos. Math. 148 (2012), doi:10.1112/S0010437X12000437
  (arXiv:1107.1846).** TeX read. B_{4,4} = 10 (improving the bound 11 of Choi-Lam-Reznick); nothing on B' in four or
  more variables.
- **Corpus records.** ulamai/UnsolvedMath, AMR-014-0019 and AMR-021-0007, both with status `open`.

## Readings
| Reading | Settled in the paper? | Note |
|---|---|---|
| FLOS convention (n = degree, m = variables), finitely many zeros | m = 4, all even n; (n,m) = (4,5) | Cor. 1.6(a) |
| Choi-Lam-Reznick / Dressler convention (variables first) | the same cases, indices swapped | e.g. their B'_{5,4} = 16 |
| Isolated real zeros of sums of squares of forms (zero set possibly infinite) | P^3 all k; quadrics on P^4 | Thms 1.4, 1.5 (statement T(k,N)) |
| Shapiro's ~#(2k,l), polynomials of degree exactly 2k | l = 3, all k; (k,l) = (2,4) | Cor. 1.6(b); T(k,l) is equivalent to the conjecture for (k,l) (Prop. 2.1) |
| Sums of squares of polynomials of degree at most k, any total degree | the same cases | Prop. 2.1(b), Cor. 1.6(c) |
| Isolated points of real algebraic sets defined by polynomials of degree <= k | 3 variables, all k (bound k^3); 4 variables, k = 2 (bound 16) | Cor. 1.6(c) |

## Results in the paper
- **Prop. 2.1, Example 2.2.** The reformulation T(k,N), its equivalence with the affine conjecture, and the grid examples
  (Choi-Lam-Reznick, Prop. 4.1), which give equality.
- **Lemmas 3.1-3.7.**
  - 3.1: weighted Bezout inequality sum_Z k^{dim Z} m_Z deg Z <= k^N, in the Stuckrad-Vogel form, with proof.
  - 3.2: multiplicities as lengths, and transversal slices.
  - 3.3: isolated real points lie in Y_0, in Sing D for real D, or in D cap conj(D).
  - 3.4: fixed components, via the gradient system.
  - 3.5: a real pencil of quadrics on P^n, n >= 3, has at most 2n isolated real points.
  - 3.6: an irreducible curve of degree a < 2w spanning P^w has at most a - w singular points.
  - 3.7: Jacobian lemma for components of multiplicity one. Its proof now states that pure dimension one of
    X = V(F_1,...,F_{N-1}) gives pure codimension j for every V(F_1,...,F_j), the hypothesis of Lemma 3.2(a).
- **Prop. 4.1.** T(k,2), the argument of Choi-Lam-Reznick, with the fixed-component step of Lemma 3.4 (needed for isolated
  zeros; the theorem of Choi-Lam-Reznick concerns finite zero sets). **Prop. 1.7** (proved in Section 4). T(2,3), hence
  B'_{4,4} = 8 and ~#(4,3) = 8.
- **Prop. 5.1.** For a curve component D in P^3: |Sing D| <= k m_D deg D (D real) and |D cap conj D| <= 2k m_D deg D
  (D not real). The proof inducts on the order of vanishing of L along D.
  - Lemma 5.2: iterated Jacobian with the Euler terms of the index-0 minors.
  - Lemma 5.3: the derivative system L^(1) satisfies m_D(L^(1)) <= m_D(L). The proof uses the chain rule and the
    valuative criterion of Lejeune-Jalabert and Teissier (I in the integral closure of m K), multiplicities of ideals
    with the same integral closure, and Northcott-Rees reductions.
- **Thm 1.4** (T(k,3)). **Remark 6.1:** the resulting bound B_{d,4} <= (d-1)^3 for all psd quaternary forms is not new; it
  also follows from the standard perturbation / critical point argument.
- **Thm 1.5** (T(2,4)). Case analysis by the dimension of the base locus:
  - a fixed component gives at most 1;
  - dim Y <= 1 gives at most 16, by Lemmas 3.6 and 3.7;
  - surface part of degree 4 forces a pencil, which gives at most 8;
  - surface part of degree <= 3 gives at most 13, using the classification of varieties of minimal degree.
- **Prop. 8.1.** T(2,5) holds when L has a fixed component, when dim Y <= 1, and when L is a pencil (at most 10 points).
  The sentence after it says what remains for B'_{4,6} = 32: T(2,5) for the systems without fixed component that are not
  pencils and have dim Y in {2,3}. For these Lemma 3.7 is not available for the curve components either, so bounds on
  all components are needed, not only on the two- and three-dimensional ones. **Remark 8.2** explains why the method
  stops.

## Computations (scripts and outputs in reproducibility/; exact unless marked heuristic)
- **Finder** (`claimant/`). All three scripts reproduce their recorded outputs exactly.
  - Lemma 5.2, iterated: 95 random cases, all as predicted.
  - m_D(G) <= m_D(F) of Lemma 5.3: 54 random examples, all for the line D = {x2 = x3 = 0}, all as predicted. 18 are
    random systems with mu = 2, 3 and 36 come from structured families with degenerate tangent cones. 42 have mu >= 2;
    the other 12 have mu = 1 (there m1 = 0) and lie outside the hypothesis of Lemma 5.3.
  - Numerical inequalities; grid counts for N <= 4 and k <= 3.
- **Lead** (`lead/lead_checks.py`). All parts pass.
  - Lemma 5.2 by an independent method: 66 cases, including the weak bound for the index-0 minors.
  - Monomial case of Lemma 5.3: 400 random ideals; e(I) >= e(K) + 3 always, with equality in 15 cases such as I = m^2.
  - All 128 rows of the case tables of Prop. 1.7, Thm 1.5 and Prop. 8.1.
  - Inequalities for k, r < 400; grids for N, k <= 4.
  - In this revision the printed labels were renumbered to the paper's numbering (Prop. 1.7, Thm 1.4, Thm 1.5 instead of
    4.2, 1.3, 1.4) and the output was regenerated; only the labels changed.
- **First independent verification run.** It checked the proofs of an earlier version by reading; it wrote no code.
- **Second independent verification run** (`independent_run_2/`, AI-assisted). Code written from the text of the paper
  before the scripts above were read; exact tests use integer data and are computed modulo two primes, which must agree.
  - Lemma 5.2: 105 cases along lines, conics, twisted cubics and nodal plane cubics; no failure.
  - Lemma 5.3, Steps 1-4: 232 systems (115 random, 117 adversarial) with 1223 pairs (F_1,F_2), and 696 checks of Step 2
    by Rees' multiplicity criterion; in every system m_D(G_1,G_2) = e(K) <= e(mK) <= e(I) <= min m_D(F_1,F_2).
  - Case mu = 1 of Prop. 5.1: the iterated Jacobian construction on 30 systems, for a real nodal cubic and a non-real
    line meeting its conjugate in a real point; no failure.
  - Numerical inequalities, case-table arithmetic of Prop. 1.7, Thm 1.5 and Prop. 8.1, Remark 8.2, Lemma 3.6(b) against
    Castelnuovo's bound, grids: all as stated.
  - Heuristic numerical counts of isolated real zeros (numpy/scipy) for (N,k) = (3,2), (3,3), (4,2), including quadrics
    on P^4 whose base locus contains a real quadric cone with only its vertex real: never more than k^N. Supporting
    evidence only.
  - It reran the four scripts above from the previous version of the package: all reproduced their recorded outputs byte
    for byte.

## First independent verification run (AI-assisted)
Verdicts of the first independent verification run (2026-09-30), for both records:

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED |
| Proofs (earlier version) | CONFIRMED: every step checked, no gap found |
| Classification | PARTIAL_PAPER (scoped partial answer) |
| Literature / novelty | no earlier proof found; no priority claim |
| Presentation | fixes required |

Required fixes and how they were applied:
1. The bound B_{d,4} <= (d-1)^3 also follows from the standard perturbation argument. Remark 6.1 says so and does not
   present it as new.
2. The question about B'_{4,4} is only a commented-out TeX line of FLOS. The paper does not claim that FLOS asked it.
3. Unified notation. The paper uses the FLOS convention throughout and translates the indices of Choi-Lam-Reznick and
   Dressler. An earlier triage remark calling "B'_{5,4} = 16" open (Dressler's indexing) is superseded by Theorem 1.5.
4. Pencils of quadrics in P^5 have at most 2n = 10 isolated real points, not 6. Prop. 8.1 states 10.
5. Read Choi-Lam-Reznick 1980 in full. Done (see above). It contains the lower bound, the ternary case, 8 <= B'_{4,4} <= 10
   and a remark without details that B'_{4,4} = 8 seems provable, but no proof of B'_{4,4} = 8. The paper credits the
   lower bound (Prop. 4.1 there) and the remark.
6. Precise references for the standard inputs. The paper cites:
   - Hartshorne I.7.2 and I.7.7;
   - Matsumura, Thms 14.3, 14.7, 17.4, 17.6 and 17.11;
   - Northcott-Rees 1954, and Huneke-Swanson Ch. 8;
   - Lejeune-Jalabert-Teissier, Thm 2.1 and Prop. 1.18;
   - Eisenbud-Harris 1987, Thm 1;
   - Fulton Sect. 12.3, Vogel 1984, and Flenner-O'Carroll-Vogel 1999, for context of Lemma 3.1.

   Castelnuovo's bound is no longer needed. The request for an expert reading of Prop. 5.1 and Lemma 5.3 stands: the note
   is released as unrefereed, and says so.
7. For AMR-021-0007:
   - joint publication with AMR-014-0019: done;
   - homogenisation step explicit: Prop. 2.1(b), and the converse in (c);
   - l = 2 attributed to Choi-Lam-Reznick;
   - (k,l) = (2,3) covered by Thm 1.4 and Prop. 1.7.

Changes relative to the version that the first run checked. The lead checks (Parts A-C) cover their case tables and the
monomial case of Lemma 5.3, and the second independent verification run (next section) checked them line by line.
- The curve case of Thm 1.5 now uses Lemmas 3.6 and 3.7, instead of Castelnuovo's bound and the arithmetic genus of
  linked curves.
- Lemma 5.3 is written out in full: the slice, the arc criterion, and the choice of a Northcott-Rees reduction.
- Prop. 1.7 has a new short proof via Lemma 3.7; the triage write-up used delta invariants.
- Prop. 8.1 proves the case dim Y <= 1 in full.

## Second independent verification run (AI-assisted)
A second AI-assisted verification run, independent of the writer, checked the version that preceded the present one
(2026-09-30). It wrote its own code from the text of the paper (`reproducibility/independent_run_2/`) before it read
the author's scripts, and then reran those. The present version differs from the checked one only by the changes listed
below.

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED: FLOS and Shapiro re-read in the TeX sources; the corpus statements match |
| Proofs | CORRECT: no mathematical error and no gap |
| Computations | CONFIRMED: own code (see Computations above); the released scripts reproduce their outputs byte for byte |
| Novelty and credit | no earlier proof found; credits accurate |
| Presentation | minor revision |
| Classification | both records partially solved |

What it checked:
- Every proof, line by line, with particular attention to the parts changed since the first run: Lemmas 3.6 and 3.7,
  Lemma 5.3 (Steps 1-4), Prop. 5.1, Prop. 1.7, the curve cases of Thm 1.5, and Prop. 8.1.
- Sources: the TeX files of FLOS, Shapiro, Dressler and Blekherman et al.; the statements cited from Choi-Lam-Reznick,
  in the GDZ scan of the journal version (Section 4, Prop. 4.1, Thm 4.6, Prop. 4.13 and the remark on p. 13); and the
  statements cited from Lejeune-Jalabert-Teissier (Thm 2.1, (iii) => (i), and Prop. 1.18), in their text.
- Attempts to break Lemma 3.6 and Lemma 5.3. Lemma 3.6(b) is sharp: for a < 2w its bound a - w equals Castelnuovo's
  bound, and a rational quartic in P^3 with one node attains it. For Lemma 5.3 the 232 test systems include adversarial
  ones and special pairs of members. No violation was found.
- Literature, with anonymous requests only: arXiv API (28 search queries, 4 id-list lookups, 4 e-prints), Crossref,
  Semantic Scholar, zbMATH Open, and one web search. OpenAlex refused anonymous requests that day; the OpenAlex citation
  lists saved when the result was found were inspected instead. Nothing proves B'_{2k,4} = k^3 for k >= 3,
  B'_{4,5} = 16, ~#(2k,l) = k^l for some l >= 3, or a sharp real Bezout bound for isolated points in R^3. As far as these
  searches show, Prop. 1.7 is the first written proof of B'_{4,4} = 8.

Required fixes and how they were applied:
1. Describe this run, and the parts changed since the first run, accurately in the Verification paragraph and in the
   release files. Done: item 3 of the Verification paragraph, this report, and `reproducibility/README.md`.
2. Stale numbering in the release package: old labels for Prop. 1.7 in this report and in the README, and for Prop. 1.7,
   Thm 1.4 and Thm 1.5 in `lead_checks.py`. Corrected; `lead_checks_output.txt` regenerated; `source.zip` rebuilt.
3. The index convention of FLOS. Section 1 now notes that their defining sentence speaks of degree m in n variables,
   while their conjecture fixes n = degree and m = number of variables, and that the paper follows the conjecture.
4. Lemma 3.7. The proof now states that pure dimension one of X = V(F_1,...,F_{N-1}) gives pure codimension j for every
   V(F_1,...,F_j), which is the hypothesis of Lemma 3.2(a).
5. The sentence after Prop. 8.1 overstated what remains for B'_{4,6} = 32. It now says that T(2,5) is needed for the
   systems without fixed component that are not pencils and have dim Y in {2,3}; that Lemma 3.7 is then not available
   for the curve components either (Thm 1.5 needed (F1), (F2) and Lemma 3.6 there); and that bounds on all components
   are needed, not only on the two- and three-dimensional ones.
6. The description of the finder's check of Lemma 5.3 (Verification, item 1). It now says that all 54 examples concern
   the line D = {x2 = x3 = 0}, that 42 of them have mu >= 2, that the 12 with mu = 1 lie outside the hypothesis of the
   lemma, and that 36 come from structured tangent-cone families.

Optional suggestions:
- Applied: the credit for l = 2 (the introduction and Table 1 now say that the theorem of Choi-Lam-Reznick concerns
  finite zero sets, and that the isolated-zero case also uses the fixed-component step, Lemma 3.4, in Prop. 4.1); the
  location of Dressler's remark (it closes Section 4 of her paper); two wording corrections in Section 3 (L is Zariski
  dense in L_C; the members in Lemma 3.1 can be chosen real when L is real).
- Not applied: a remark relating Step 2 of Lemma 5.3 to the classical fact that f lies in the integral closure of
  m J(f); a mention of Kharlamov's theorem behind B_{4,4} = 10; a reference for Castelnuovo's bound in the Verification
  paragraph; highlighting Cor. 1.6(c) in the introduction; and a note that m >= 2 in Conjecture 1.1, which is already
  stated for m >= 2. None of them affects correctness or credit.

## Relation to the literature, novelty and scope
- **Searches (September 2026).**
  - arXiv API: 74 search queries (30 when the result was found, 16 in the first verification run, 28 in the second) on
    real or isolated zeros of sums of squares and nonnegative forms, quaternary forms, Choi-Lam-Reznick,
    Ottaviani-Shapiro, base loci, and isolated real solutions versus Bezout.
  - OpenAlex: works citing Choi-Lam-Reznick (64), FLOS (7), Shapiro (10), Dressler (3) and BHORS (26).
  - Semantic Scholar: works citing Choi-Lam-Reznick (97), Shapiro (4) and Barone-Basu (22).
  - zbMATH Open, including the 41 documents citing Choi-Lam-Reznick; Crossref (every DOI in the paper was verified
    there, again in the second verification run).
  - Four web searches in total, the last one in the second verification run.
  - Nothing gives B'_{n,4} for n >= 6, B'_{4,5}, or ~#(2k,l) for l >= 3.
  - Barone-Basu (Proc. LMS 112 (2016)) note that the Bezout inequality fails over the reals and prove a weak real
    analogue for connected components.
  - A web search found no general real Bezout bound for isolated points.
- **Caveats.**
  - We do not know Dressler's source for B'_{4,4} = 8.
  - Literature in other languages was not checked systematically.
  - This negative search is not a proof of priority.
- **Scope.** Settled here: B'_{2k,4} = k^3 (all k), B'_{4,5} = 16, ~#(2k,3) = k^3 (all k), ~#(4,4) = 16, and a short
  proof of B'_{4,4} = 8. Known before: m <= 3 (resp. l <= 2), and the lower bounds. Open: all other cases with m >= 5 and
  n >= 4 (resp. l >= 4 and k >= 2). Partial: B'_{4,6} = 32 for systems with dim Y <= 1, pencils, or a fixed component.
  Suggested corpus status for both records: partially_solved.

## 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/