# Verification report — OWR-14299088-013 (two multi-affine polynomials, Basu–Perrucci)

Verification date: 2026-09-30 (revised the same day after the paper-stage referee report).

**Verdict.** The answer is no. The number of connected components of the common real zero set of two multi-affine
polynomials of degree at most d cannot be bounded in terms of d alone, independently of the number n of variables,
for any d ≥ 4. For m ≥ 2 the multi-affine polynomials P_m = x_0 − Σ x_i y_i (degree 2) and
Q_m = x_0 Σ x_i y_i − 2 Σ_{i<j} x_i y_i x_j y_j − 2 Σ x_i y_i + m (degree 4) in n = 2m + 1 variables satisfy
Q_m = (Σ x_i y_i) P_m + Σ (x_i y_i − 1)^2. So their common real zero set is {x_0 = m} × (m hyperbolas), and it has exactly
2^m components. For m = 1 the same formulas give degrees (2, 3) and 2 components. The question remains open for degree
bounds d = 2 and d = 3; d = 1 is trivial. The note is unrefereed.

## Statement checked
- **Primary source.** S. Basu, D. Perrucci, "Topology of real multi-affine hypersurfaces and a homological stability
  property", Adv. Math. 420 (2023) 108982, doi:10.1016/j.aim.2023.108982; arXiv:2204.01595 (only v1). The arXiv text
  was read. The question appears twice:
  - in the sentence right after Example 2.2 (p. 6), where the authors say they do not know whether a bound exists for
    two polynomials of degree at most d;
  - as open problem 2 of Section 4 (p. 24), which asks for a bound independent of n.
- **Restatement.** Oberwolfach Report 9/2025 (MFO–RIMS Tandem Workshop "Optimization, Theoretical Computer Science and
  Algebraic Geometry: Convexity and Beyond", February 2025), Oberwolfach Rep. 22 (2025), no. 1, 405–450,
  doi:10.4171/OWR/2025/9. The open problem, contributed by S. Basu (joint work with D. Perrucci), is on pp. 445–446.
  - The problem is the same as in [BP].
  - The report recalls the one-polynomial bound 2^(d−1) and the three-polynomial example with C(n,k) components.
- **Definition.** "Multi-affine" means deg_{X_i} P ≤ 1 for every i ([BP] Definition 1.2); the construction uses this
  definition.
- **Corpus record.** ulamai/UnsolvedMath, OWR-14299088-013 (status `open`). Its statement, source and DOI match the
  report.
- **Paper-stage check.** The paper-stage referee (referee 2) fetched OWR 9/2025 anonymously through its DOI and read
  [BP] arXiv v1 in full. The OWR item runs from p. 445 to p. 446, with the problem statement on p. 446, and every
  numbered citation of [BP] in the paper matches the arXiv numbering. Verdict: CONFIRMED.

## Readings
| Reading | Refuted? | Witness |
|---|---|---|
| b_0(Z(P,Q)) ≤ f(d) for all n and all multi-affine P, Q of degree ≤ d ([BP] §4 problem 2; OWR 9/2025) | yes, for every d ≥ 4 | (P_m, Q_m), m ≥ 2: degrees (2,4), n = 2m+1, b_0 = 2^m; for even n add an unused variable |
| the same with larger degree bounds, e.g. degrees (d, 2d) | yes | Corollary 3.3: f_j = z_{j,1}⋯z_{j,d} − 1, degrees (d, 2d), n = dk+1, b_0 = 2^((d−1)k) |
| bounded version: components of Z(P,Q) ∩ [−R,R]^n | yes | Proposition 4.1(a): exactly 2^m components for R ≥ m |
| bounded version for any sequence of sets B_n with nonempty interior (e.g. compact convex bodies) | yes | Proposition 4.1(b): rescaled (P_m, Q_m) give ≥ 2^m components in B_n |
| over an arbitrary real closed field (the setting of [BP]) | yes | the same polynomials; Remark 4.2: semi-algebraic homeomorphisms, semi-algebraically connected components, B semi-algebraic in Proposition 4.1(b) |
| degree bound d = 1 | no (the bound exists) | two affine functions: the zero set is an affine subspace or empty |
| degree bounds d = 2, 3 | open | Proposition 5.5 excludes only pairs of "elimination type" (at most 2 components) |

## Results in the paper
- **Theorem 1.2.** For m ≥ 1:
  - P_m and Q_m are multi-affine, deg P_m = 2, deg Q_m = 4 for m ≥ 2 (3 for m = 1);
  - the ideal-membership certificate Q_m = T·P_m + Σ (x_i y_i − 1)^2 holds (T = Σ x_i y_i);
  - Z(P_m, Q_m) = {x_0 = m, x_i y_i = 1 ∀i} has exactly 2^m components, each homeomorphic to R^m.
- **Corollary 1.3.** For every n ≥ 5 there is a pair of degrees (2, 4) with 2^⌊(n−1)/2⌋ components. So the problem
  has a negative answer.
- **Lemma 3.1 (block lemma).** For multi-affine f_1..f_k of degree ≤ d in disjoint variable sets, the polynomials
  P = x_0 − S and Q = x_0 S − 2 Σ_{j<l} f_j f_l (S = Σ f_j) are multi-affine of degrees ≤ d and ≤ 2d. They satisfy
  Q = S·P + Σ f_j^2, and Z(P,Q) ≅ Z(f_1) × ⋯ × Z(f_k).
- **Remark 3.2.** Theorem 1.2 is Lemma 3.1 with f_i = x_i y_i − 1, up to the shift x_0 → x_0 − m and the addition of
  −m·P_m.
- **Corollary 3.3.** Degrees (d, 2d) and 2^((d−1)k) components in dk + 1 variables.
- **Remark 3.4.** The construction is the two-polynomial analogue of [BP] Example 2.2.
  - There, P_1 and P_2 fix the power sums N_1 and N_2, so the sum of squares Σ X_i^2(X_i − 1)^2 is congruent to the
    multi-affine P_3 modulo (P_1, P_2).
  - Here one auxiliary-variable equation does the same job.
- **Proposition 4.1.** The bounded versions: exactly 2^m components in [−R,R]^n for R ≥ m, and at least 2^m inside any
  set with nonempty interior after a coordinatewise affine rescaling.
- **Remark 4.2.**
  - It relates Proposition 4.1(b) to [BP] Proposition 1.1, whose hypothesis is a compact convex semi-algebraic set of
    dimension n.
  - It states in which form Theorem 1.2, Corollary 1.3 and Sections 2–4 hold over any real closed field K:
    semi-algebraic homeomorphisms, semi-algebraically connected components, and B semi-algebraic in
    Proposition 4.1(b).
  - The justification: the identities are algebraic; a sum of squares in K vanishes only if every term does;
    (0,∞) ≅ K via a ↦ a − 1/a; intervals and finite products of semi-algebraically connected sets are semi-algebraically
    connected ([BCR98] §§2.4–2.5).
- **Proposition 5.5 (limits of the method).** Consider pairs of elimination type, P = x_0 − T and Q = C·P + Σ h_j^2.
  - (a) If deg T ≤ 1 or deg C ≤ 1, then the zero set is empty or an affine space.
  - (b) If both degrees are ≤ 3, the zero set has at most 2 components; this is attained for m = 1.
  - The tools are a support lemma (Reznick's Newton-polytope argument, Lemma 5.2), an elementary divisibility lemma for
    indefinite quadratic forms (Lemma 5.3) and the affine normal forms of quadrics (Lemma 5.4).
  - This does **not** settle degree bounds 2 and 3 for general pairs.
- **Section 6, question 2.** For every degree bound d ≥ 4, Corollary 3.3 with blocks of degree d' = ⌊d/2⌋ gives
  2^((d'−1)k) = (2^(1−1/d'))^(n−1) components for n = d'k + 1, k ≥ 2.

## Computations (exact unless stated; scripts and outputs in reproducibility/)
- **Finder** (`claimant/verify_construction.py`, standard library, exact rational arithmetic). It checks:
  - Theorem 1.2(a) and the identity Q_m(x_0 = T) = Σ(x_i y_i − 1)^2 for m ≤ 12. This is equivalent to the
    certificate, because both sides have x_0-coefficient T.
  - Lemma 3.1(a),(b) only, for f_j = z_{j,1}⋯z_{j,d} − 1 with 2 ≤ d ≤ 4 and 1 ≤ k ≤ 6: multi-affinity, the degrees, and
    Q|_{x_0=S} = Σ f_j^2 (equivalent to Lemma 3.1(b), for the same reason).
  - Exact rational zeros in all 2^m sign classes for m ≤ 10.
  - Result: `ALL CHECKS PASSED`.
- **Finder, heuristic** (`claimant/numeric_components.py`, numpy/scipy). It uses sampling and clustering and does not
  use the identity.
  - For m = 1, 2 it finds 2 and 4 clusters.
  - For m = 3 it observes all 8 sign vectors, but its count is 9 clusters: this is sampling noise.
  - It is not used in the paper.
  - The recorded output of the original run ended with a shell `time` line, which the script does not print. That
    line was removed from the release copy.
- **Independent referee** (`referee/indep_check.py`, written from scratch). It checks:
  - the exact certificate for m ≤ 10;
  - exact integer evaluation at 200 random points each for m ∈ {11, 15, 20, 25, 30, 40};
  - the block-lemma certificate for f_j = z_{j,1}⋯z_{j,d} − 1 with 2 ≤ d ≤ 5 and k ≤ 5;
  - the count 2^(d−1);
  - numerically, a least-squares refutation attempt for m = 2, 3, 4. Every zero found had x_i y_i = 1 and x_0 = m
    within 2·10^−5, and all 2^m sign patterns occurred.
  - Result: `OVERALL: PASS`.

  The referee's re-run of the finder's script is byte-identical to the finder's output.
- **Lead** (`lead/lead_check.py`, written for the paper, standard library, exact). It checks:
  - the certificate for m ≤ 16;
  - Remark 3.2 for m ≤ 12;
  - Lemma 3.1(a),(b) on 200 random multi-affine block instances;
  - the certificate and the exact degrees (d, 2d) of Corollary 3.3 for 2 ≤ d ≤ 5 and 2 ≤ k ≤ 5 (the component count
    2^((d−1)k) is only printed as a prediction, not computed);
  - the rescaling of Proposition 4.1(b) for m ≤ 5;
  - Remark 5.6.
  - Result: `ALL LEAD CHECKS PASSED`.
- **Paper-stage referee** (`referee2/r2_check.py`, written before any other script was read, standard library, exact
  `Fraction` arithmetic, under 10 s). It checks:
  - Theorem 1.2(a),(b) for m ≤ 16, including the abstract's formula for Q, the coefficient −2 of x_1 y_1 x_2 y_2 and the
    form of Q_1;
  - an exact common zero in every one of the 2^m sign classes for m ≤ 10;
  - Lemma 3.1(a),(b) on 250 random multi-affine block instances;
  - Corollary 3.3 for 2 ≤ d ≤ 5 and 2 ≤ k ≤ 5 with n = dk + 1 ≤ 21: the exact degrees (d, 2d), and an exact common zero
    in each of the 2^((d−1)k) sign classes whenever 2^((d−1)k) ≤ 4096;
  - Remark 3.2 for m ≤ 12, and the Newton-identity computation of Remark 3.4 for k ≤ 11;
  - the box criterion of Proposition 4.1(a) (m ≤ 4) and the rescaling of Proposition 4.1(b) (m ≤ 5);
  - Lemma 5.2 on 276 random instances (the script prints "300", but it skips the 24 draws in which all h_j vanish; an
    instrumented copy with the same seed gives 276);
  - the indefiniteness step T_2(e_a ± e_b) = ±c of Proposition 5.5(b), on 250 random forms.
  - Result: `ALL EXACT CHECKS PASSED`.
- **Paper-stage referee, numerical** (`referee2/r2_numeric.py`, numpy/scipy), evidence only.
  - A least-squares refutation attempt for m = 2..5, with 4000 random starts each. Every converged zero satisfies
    |x_i y_i − 1|, |x_0 − m| ≤ 7.6·10^−6, and all 2^m sign patterns occur.
  - A cluster count of the grid cells of [−3,3]^(2m) with |x_i y_i − 1| < ε for all i. It finds 2 clusters for m = 1 and
    4 clusters of equal size for m = 2. It checks only the number of components of {x_i y_i = 1}, not the description
    of the zero set.
- **Re-runs.** The four exact scripts (finder, independent referee, lead, paper-stage referee) were re-run from a clean
  copy of `reproducibility/` on 2026-09-30, with byte-identical outputs. `r2_numeric.py` was also re-run and
  reproduced its recorded output byte-identically. Referee 2 had earlier re-run the finder's, the independent
  referee's and the lead's scripts (byte-identical) and the finder's heuristic script (same output).

## Independent adversarial audit
Two independent verifiers, 2026-09-29:

| Item | Verifier 1 | Verifier 2 |
|---|---|---|
| Classification | PAPER_CANDIDATE | PAPER_CANDIDATE |
| Correct | yes | yes |
| Answers the question as intended | yes | yes |
| Statement fidelity (BP arXiv text and OWR 9/2025 text read) | CONFIRMED | CONFIRMED |
| Proofs (construction, block lemma, box variant, degree-3 analysis) | CONFIRMED | CONFIRMED |
| Computations | CONFIRMED (own exact and numerical code) | CONFIRMED (own exact code for m ≤ 7; finder's script re-run) |
| Novelty | nothing found; keep claim modest | nothing found; keep claim modest |

All required fixes were applied.
- **Verifier 1.**
  1. The block-lemma remark now reads "up to a shift of x_0 and the addition of a multiple of P" (Remark 3.2).
  2. The ideal-membership certificates are stated and used (1), and so is Lemma 3.1(b).
  3. The family is taken with m ≥ 2 for degree 4.
  4. Translation/scaling invariance and [BP] Proposition 1.1 are discussed, and the bounded version is proved for any
     sets with nonempty interior.
  5. The novelty wording is modest, and both posing sources are cited.
- **Verifier 2.**
  1. The note presents the result as the two-polynomial analogue of [BP] Example 2.2 (Remark 3.4).
  2. The literature log records Semantic Scholar's single unrelated citing record and the repeated OpenAlex HTTP 429.
  3. The degree wording is uniform: (2,4) for m ≥ 2, (2,3) for m = 1.
  4. The m = 3 heuristic count has been dropped from the paper and is labelled sampling noise here.
  5. Both citation places are given ([BP] remark after Example 2.2 and §4 problem 2; OWR 9/2025 pp. 445–446, DOI
     10.4171/OWR/2025/9).

At the paper stage the degree-3 analysis was rewritten in self-contained form. The real Nullstellensatz and
sign-changing criterion were replaced by the elementary Lemma 5.3, and the conclusion is unchanged. The two verifiers
did not see these rewritten proofs (Lemmas 5.2–5.4). They were checked by the author and then re-derived by hand by
the paper-stage referee (below), who found them correct.

### Paper-stage referee (referee 2, 2026-09-30)
Verdict: **correct; release as an unrefereed note after minor fixes; not fatal.** The referee
- read OWR 9/2025 and [BP] in the original;
- re-derived all 16 numbered statements and remarks by hand, including the rewritten Lemmas 5.2–5.4;
- wrote its own exact and numerical code before reading any other script;
- re-ran all packaged scripts.

| Item | Verdict |
|---|---|
| Statement fidelity (OWR 9/2025 and [BP] read in the original) | CONFIRMED |
| Proofs (all numbered statements and remarks, line by line) | CONFIRMED; only minor precision points |
| Computations (own code written first; all packaged scripts re-run) | CONFIRMED |
| Novelty | not found in the literature; no prior or concurrent resolution; the OpenAlex `cites:` list is still unavailable |
| Presentation / house style / release package | compliant, with minor fixes |
| Fatal | no |

Required fixes:
1. **Remark 4.2, real closed fields.** The statement is now precise.
   - It uses semi-algebraic homeomorphisms and semi-algebraically connected components, and requires B to be
     semi-algebraic in Proposition 4.1(b).
   - The justification: the identities are algebraic; sums of squares in K vanish only termwise; (0,∞) ≅ K via
     a ↦ a − 1/a; intervals and finite products of semi-algebraically connected sets are semi-algebraically connected
     ([BCR98] §§2.4–2.5); this gives Lemma 2.2 over K.
   - "Verbatim" was dropped, also from the sentence on [BP] Proposition 1.1.
2. **Remark 4.2.** [BP] Proposition 1.1 is now quoted for a compact convex semi-algebraic set of dimension n.
3. **Proposition 4.1(b).** The proof applies Lemma 2.1(ii) to φ^{−1}(Y) = (m/r)(Y − c).
4. **Verification, item 2.** The finder's Lemma 3.1 check is for f_j = z_{j,1}⋯z_{j,d} − 1 with 2 ≤ d ≤ 4 and
   1 ≤ k ≤ 6, parts (a),(b) only.
5. **Verification, item 3.** The family and the range 2 ≤ d ≤ 5, k ≤ 5 are named.
6. **Verification, item 4.** It now reads "the certificate and the exact degrees (d, 2d) of Corollary 3.3" for
   2 ≤ d ≤ 5, 2 ≤ k ≤ 5, and it says that the component count is not computed.
7. **"Involve no computation".** The sentence now reads "The proofs in Sections 4 and 5 do not use computation."
8. **The paper-stage audit is recorded.**
   - The paper has a new Verification item 6.
   - `r2_check.py`, `r2_numeric.py` and their outputs are in `reproducibility/referee2/`, with README rows. This is
     the folder name used in this round; the report proposed `referee/`, which holds the first verifier's code.
   - The "not re-audited" sentence above was updated.
9. **Scope and priority.**
   - It now reads "Theorem 1.2 and Proposition 4.1 answer …".
   - The search record states that zbMATH lists no citing document, that arXiv was re-searched on 2026-09-30, and
     that the OpenAlex citing-works list was again rate-limited on 2026-09-30.
10. **OpenAlex retry.** Not yet possible: see Caveats below. The anonymous budget resets at 00:00 UTC, and this
    revision was made before the reset. The caveat is kept.
11. **zbMATH.** `rf:1511.14033` returned the API's empty-result answer, not a failure. This is corrected here (below)
    and in RESULT.md §5.
12. **Shell timing line.** The trailing shell `time` line was removed from
    `reproducibility/claimant/numeric_components_output.txt`.

The optional suggestions were also applied.
- (13) The basis vectors in the proof of Proposition 5.5(b) are now ε_a, so they no longer clash with e_2.
- (14) γ now denotes only elements of supp(G); the exponents of T and C are β, β′, and the constant term in Lemma 5.4
  is δ.
- (15) The pairs of Lemma 3.1 are of elimination type with T = C = S.
- (16) "x_0T = Σ x_0t_i acts as a proxy for T²", and "for d = 1 a bound exists trivially, namely f(1) = 1".
- (17) The searched sources in the scope paragraph are written as prose.
- Section 6, question 2 now covers odd degree bounds, with blocks of degree ⌊d/2⌋.

After the edits:
- The paper was rebuilt with tectonic: no errors, no overfull or underfull boxes, 10 pages. All pages were rendered at
  1.4× and inspected.
- main.tex, references.bib and paper.pdf were copied into release/, and source.zip was rebuilt. The zenodo/ copies and
  their sha256, size and md5 were refreshed.
- The abstract is unchanged, so the Zenodo description is unchanged.
- The pre-revision source is preserved as `paper/main_v1_prereferee_2026-09-30.tex`.

## Relation to the literature, novelty and scope
- **Searches (September 2026, anonymous requests, logged).**
  - arXiv API: abstracts/titles with "multi-affine"/"multiaffine" (85 hits up to 2026-09-24), combined with
    "connected components", "Betti" and "zero set"; all papers of Perrucci; Basu's recent papers up to 2608.07794.
  - arXiv again on 2026-09-30, local time (referee 2; 2026-09-29, about 22:05 UTC, in queries.log).
    - 86 abstract/title hits for "multi-affine"/"multiaffine", newest 2609.30587; none is on real zero sets or
      components.
    - The synonym searches ("multilinear" with "connected components" and "real"; "two multi-affine"/"two multilinear
      polynomials") found nothing relevant, and the author feeds show no follow-up.
    - Basu's later papers 2411.11729 and 2509.14079, read in full, do not mention multi-affine polynomials.
  - zbMATH: both authors, 2023–2026; titles with "multi-affine", 2023–2026: nothing relevant. The citation search
    `rf:1511.14033` returned the API's empty-result answer (status 404, "successful access. No results found."), on
    2026-09-29 and again on 2026-09-30. It is not a failure: zbMATH indexes no document citing [BP]. Only the auxiliary
    query `ci:1511.14033` failed (HTTP 502).
  - Crossref: DOI confirmed; is-referenced-by-count 0 (checked again by referee 2); bibliographic query found nothing
    relevant.
  - Semantic Scholar: one citing record of arXiv:2204.01595, an unrelated 2021 entry "Polynomial Ideals".
  - OpenAlex: the single-work record gives cited_by_count 0 (paper stage and referee 2). Every list query (topic
    search, and the `cites:` filter at the paper stage and again for referee 2 on 2026-09-30) returned HTTP 429: the
    anonymous daily budget was exhausted, and it resets at midnight UTC.
  - Web search (one call per agent, five in total): only [BP], the unrelated 2305.07403 (real zero amalgamation),
    Basu's CV and unrelated fewnomial or sparse-polynomial work.
  - Repository (referee 2): no competing or concurrent write-up of this problem.
  - Nothing resolves the question or contains the construction.
- **Caveats.**
  - The OpenAlex citation list could not be retrieved; the last attempt was on 2026-09-30 (local time). **Retry
    `https://api.openalex.org/works?filter=cites:W4361223843` after the anonymous quota resets (00:00 UTC) and before
    any public release, and examine any citing work.** If the retry fails again, keep this caveat and the sentence in
    the paper.
  - The construction is elementary and in the spirit of [BP] Example 2.2, so it may be known to specialists.
  - This negative search is not a proof of priority.
- **Scope.**
  - The note answers [BP] §4 problem 2 and the OWR 9/2025 problem negatively for every degree bound d ≥ 4:
    - in R^n (Theorem 1.2);
    - inside boxes and inside any sets with nonempty interior (Proposition 4.1);
    - over real closed fields, with semi-algebraically connected components (Remark 4.2).
  - Degree bounds 2 and 3 remain open; Proposition 5.5 excludes only the elimination-plus-sum-of-squares mechanism.
- **Suggested corpus status.** Change OWR-14299088-013 from `open` to `solved` (negative answer). The explicit family has
  degrees (2,4) and 2^m components in R^(2m+1).

## 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/