# Verification report — AMR-022-4023 (Hayman–Lingham, Problem 4.23 (P. Erdős): connected lemniscates in R^m)

Verification date: 2026-10-16.

**Verdict.** The note answers one of the two questions of the record, with complete proofs.
- **Settled.** If E = {x ∈ R^m : ∏|x − x_i| ≤ 1} is connected, then |x − c| < 2 for every x ∈ E, where c is the
  centroid of the points; this holds for all m ≥ 1 and n ≥ 1 (Theorem 1.1). So the covering question of Problem
  4.23 has the answer yes, in every dimension, with the centroid as centre.
- **Sharpness.** Collinear Chebyshev configurations give connected sets with max_{x∈E}|x − c| = 2·2^(−1/n) and
  diameter 4·2^(−1/n) (Proposition 1.2); so the constant 2 cannot be lowered uniformly in n, already for m = 1.
- **Fixed n.** The best radius about the centroid, R_m(n), does not depend on the dimension: R_m(n) = R_1(n), and
  2·2^(−1/n) ≤ R_1(n) ≤ 2 (Corollary 1.3).
- **Open.** Whether R_1(n) = 2·2^(−1/n) for n ≥ 3 (Question 1.4; true for n ≤ 2; numerical evidence only, n ≤ 12);
  the volume question of Problem 4.23.
- **Credit.** The question is Erdős's; the planar case is Pommerenke's theorem (Michigan Math. J. 6 (1959),
  Theorem 3), which the note proves again in the closed-set form needed. The note's contribution is the projection
  reduction, which gives the covering statement in every dimension and the dimension-independence of the best radius
  about the centroid. The sharpness example (the zeros of a Chebyshev polynomial on a line) is proved in full but is
  not claimed as new for m ≤ 2.

The note is unrefereed. Suggested record status: **partially solved** (covering question answered yes in every
dimension; volume question open).

## Statement checked
- **Primary source.** W. K. Hayman, E. F. Lingham, *Research Problems in Function Theory (New Edition)*,
  arXiv:1809.07200v2 (21 September 2018), read in a local copy of the v2 PDF (text extracted with pypdf):
  Problem 4.23 and Update 4.23 are on PDF page 80 = printed page 79. The heading before Problem 4.7 (PDF p. 75)
  defines E_f^(n) = {|f| ≤ 1} for monic f of degree n; Problems 4.7–4.12 and their updates were read. The published
  book (Springer 2019, doi:10.1007/978-3-030-25165-9) was not consulted.
  - Problem 4.23: points x_1, …, x_n in R^m and E_n^(m) = {∏|x − x_i| ≤ 1}; (a) when is the maximum volume attained
    and how large can it be (Piranian: the ball is not extremal for m = 3, n = 2); (b) if E is connected, can it be
    covered by a ball of radius 2; for m = 2 this was proved by Pommerenke [640] = Michigan Math. J. 6 (1959)
    221–225. Attributed to P. Erdős. Update 4.23 reports that no progress had been reported.
- **Origin.** erdosproblems.com #509 (last edited 29 December 2025, read 16 October 2026) states that Erdős asked
  the higher-dimensional generalisation as Problem 4.23 in Hayman's collection of 1974 (Symposium on Complex
  Analysis, Canterbury 1973, LMS Lecture Note Ser. 12, pp. 155–180, doi:10.1017/CBO9780511662263.034; not seen).
  Table 2 of the arXiv edition (PDF p. 254) lists Problems 4.22–4.24 among the problems proposed in that collection
  (document B of its Table 1, pp. 143–180); chapter 4 of the 1967 book has Problems 4.1–4.21.
  Erdős's list of 1961 (pp. 246–247) contains the planar lemniscate problems but not the question in R^m.
- **Corpus record.** ulamai/UnsolvedMath 1.6.0, AMR-022-4023 ("Research Problems in Function Theory — Problem
  4.23", upstream status open). Its text agrees with the source except that the citation label "[640]" after
  "Pommerenke" is missing.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Covering question, ball with any centre | yes, for all m, n (even with centre c and strict inequality) | Theorem 1.1 |
| Ball centred at the centroid | yes, open ball B(c, 2) | Theorem 1.1 |
| Ball centred at one of the points x_i | not the source's reading; false for large n (collinear Chebyshev points) | — |
| "Ball of radius 2" closed or open | irrelevant: the inequality is strict | Theorem 1.1 |
| Higher dimensions read as C^k with the Hermitian norm | the case R^(2k), covered | Theorem 1.1 |
| Lemniscates of polynomials in several complex variables | a different object (Cuyt–Driver–Lubinsky 1996), not treated | Scope and priority |
| Disconnected E | the question does not concern it; two points at distance d > 2 give a disconnected E of diameter (d² + 4)^(1/2) | Section 1.1 |
| Best constant for fixed n | the same in all dimensions; its value is open for n ≥ 3 | Corollary 1.3, Question 1.4 |
| Volume question | not treated | Section 5 |

## Results in the paper
- **Lemma 2.1.** If a connected J ⊆ L(q) = {|q| ≤ 1} contains all zeros of the monic polynomial q, then L(q) is
  connected (every component of {|q| < r} contains a zero, by the minimum modulus principle; nested compact
  connected sets).
- **Lemma 2.2 (Pommerenke's theorem, closed-set form, strict).** If L(q) is connected, then |z − c_q| < 2 on L(q),
  c_q the mean of the zeros. Proof: exterior Riemann map ψ with q∘ψ = w^n, constant term = c_q, Bieberbach's
  |a_2| ≤ 2 with its equality case. Remark 2.3: Pommerenke's Theorem 3 assumes the open set {|f| < 1} connected;
  the closed form is needed because the projected polynomial may have critical points on |q| = 1 (Chebyshev).
- **Theorem 1.1.** Projection π(y) = ⟨y − c, u⟩, u = (x − c)/|x − c|; Cauchy–Schwarz gives π(E) ⊆ L(q) ∩ R for the
  real-rooted q with zeros π(x_i), mean 0; π(E) is an interval containing all zeros; Lemmas 2.1 and 2.2.
- **Remarks 3.1–3.4.** Connected subsets of E containing all x_i lie in B(c, 2); the same for every norm N
  (norming functional), radius in N; in R^3 a component of E can contain none of the x_i (six points ±a e_k with
  27/32 < a^6 ≤ 1, proved); diam E ≤ 4·2^(−1/n) by Chebyshev's theorem, and Jung's theorem alone would not give
  radius 2.
- **Proposition 1.2.** q_n(t) = T_n(t/(2ρ)), ρ = 2^(−1/n), is monic with the zeros t_k; E is the body of
  revolution of L(q_n); |T_n(w)| ≤ 1 implies |w| ≤ 1 (|cos(α+iβ)|² = cos²α + sinh²β, |sinh nβ| ≥ n|sinh β|,
  |sin nα| ≤ n|sin α|); connectedness by radial monotonicity.
- **Corollary 1.3.** Embedding a line gives R_m(n) ≥ R_1(n); the projection gives R_m(n) ≤ R_1(n).
- **Lemma 6.1.** If every closed ball with diameter [p, x_i] lies in E, then E is star-shaped with respect to p
  (used only as the connectedness certificate of the program of the note).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **Program of the note** (`check_note.py`, about 35 s; all checks passed): (i) exact (SymPy) identities of the
  proof of Proposition 1.2 for n ≤ 12 (recurrences, monicity, mean 0, T_n(±1), Pell identity, T_n' = nU_{n−1},
  Sturm counts, |cos(α+iβ)|², sin nα = sin α U_{n−1}(cos α), |U_{n−1}| ≤ n, Taylor coefficients), and numerically
  max |w| = 1 on the curve |T_n(w)| = 1; (ii) 87 configurations in R^2, R^3, R^4 certified connected by Lemma 6.1
  (verified with a branch-and-bound covering of the spheres), R(c) by ray casting, the projection inequality for
  every sampled direction, two controls (largest radii 1.3643, 1.3324, 1.2988; no violation); (iii) Lemma 2.2 for
  z^n − 1 exactly and seven explicit polynomials numerically (largest 1.8353); (iv) the n = 2 formula and the
  identities of Remark 3.3 exactly.
- **First computation** (`original/`): exact and numerical checks of Proposition 1.2 for n ≤ 12; 2000 random
  configurations in R^1–R^6 (1471 accepted; largest R(c) 1.6086); near-extremal collinear configurations
  (R(c) = 2·2^(−1/n) to 6 decimals); Lemma 2.1 by grid labelling and Lemma 2.2 on random lemniscates (largest
  1.8448). Its connectedness test (sampled segment graph) shows only that the x_i and the sampled points lie in one
  component.

## Independent verification runs
Two independent verification runs, both AI-assisted, examined the first written account of the results; each
wrote its own programs, and neither read the other's report. Run B wrote its programs from the definitions before
reading those of the first computation.

| Item | Run A | Run B |
|---|---|---|
| Statement against the source (arXiv v2), the record, readings | CONFIRMED (page: PDF 80 = printed 79; record drops "[640]") | CONFIRMED |
| Theorem 1.1 (projection, all m, n) | CONFIRMED (re-derived, degenerate cases) | CONFIRMED (re-derived) |
| Lemma 2.1 | CONFIRMED | CONFIRMED (second route via the critical-value criterion) |
| Lemma 2.2 (strict, centroid) | CONFIRMED (Koebe equality case) | CONFIRMED |
| Proposition 1.2, lower bound | CONFIRMED; exact/60-digit for n ≤ 20 | CONFIRMED; exact certificates for n ≤ 40 |
| Proposition 1.2, upper bound max|x − c| ≤ 2·2^(−1/n) | PROVED (first account had it numerically only) | PROVED independently |
| Corollary 1.3 (R_m(n) = R_1(n)) | — | FOUND and proved |
| Remark 3.1 (connected subsets containing all x_i) | — | FOUND and proved |
| Remark 3.2 (all norms, norming functional) | FOUND | FOUND; ℓ∞ example computed |
| Remark 3.3 (component without x_i in R^3) | — | found numerically (six-point example); proof added at the writing stage |
| Remark 3.4 (diameter bound; Jung's theorem insufficient) | FOUND | — |
| Numerical tests of Theorem 1.1 | 1400 configurations in R^3, R^4 (voxel labelling), largest R(c) 1.6742 / 1.6399, none ≥ 2 | planar adversary, grid labelling in R^3, hill climbs; largest 1.8001 (n = 8) < 2·2^(−1/8); wrong inputs detected |
| Question 1.4 evidence | largest ratio R(c)/(2·2^(−1/n)) 0.99994 | SLSQP over real configurations, n ≤ 12: maximum 2·2^(−1/n) within 1.7·10^(−10); complex zeros (n ≤ 8) end at real Chebyshev configurations |
| Re-run of the first computation's programs | Chebyshev program: identical output | all five programs: byte-identical outputs |
| Literature and novelty | not found in print; JMR 2016 found | not found in print; Erdős–Hwang 1978 found |
| Record level | PARTIAL | PARTIAL |

Neither run found a mathematical error or a gap.

**Corrections required by runs A and B**, all applied in the note (details in the problem folder's audit file):
1. Pages: PDF p. 80 / printed p. 79 of arXiv:1809.07200v2 (2018); the 2019 book was not read (A-F6, B-1).
2. Erdős 1961: pp. 246–247 (A-F6, B-2).
3. Origin: Hayman's 1974 collection, as stated on erdosproblems.com #509 (B-3).
4. No content attributions to unread papers (Pommerenke's real-zeros paper, Erdős–Herzog–Piranian 1958); the fact
   that every component of {|q| < r} contains a zero is proved in Lemma 2.1, not attributed (A-F1).
5. erdosproblems #1046 is stated for the open set; Lemma 2.2 is the closed-set form; no non-verbatim quotation
   (A-F2).
6. The unproved remark on coplanar configurations was removed (A-F3).
7. Proposition 1.2 now includes the upper bound (exact value 2·2^(−1/n), diameter 4·2^(−1/n)) with proof (A-F4).
8. Erdős–Hwang (Aequationes Math. 17 (1978) 344–347) and Jahn–Martini–Richter (J. Convex Anal. 23 (2016) 733–774)
   are cited as related work without covering results (A-F5, B-7).
9. Numerical connectedness tests are described as showing only that the x_i and the sampled points lie in one
   component (B-4); the exact and numerical parts of the first Chebyshev program are distinguished, and the Pell
   identity is cited (B-5); the remark on nearly collinear extremal configurations is labelled an observation
   (B-6).
10. erdosproblems #509 "mentions the question without a result" (A-F6); the record text "agrees with the source
    except the dropped label [640]" (A-F6).
11. Optional O1 (all norms via a norming functional): Remark 3.2.

**Changes made when the note was written, after runs A and B.**
- Pommerenke (1959) and Cuyt–Driver–Lubinsky (1996) were read (from copies obtained through a browser; the
  publishers' sites refused automated requests). Remark 2.3 states Pommerenke's Theorem 3 exactly (open set
  {|f| < 1} connected ⇒ the curve |f| = 1 lies in |z − ζ| < 2) and explains why the closed-set form is needed; it
  derives the non-strict closed form from his theorem by scaling, and records his criterion (p. 222).
- Lemma 6.1 (a certificate of star-shapedness) and the program `check_note.py` were written; the proof of the
  example in Remark 3.3 was added (the example had been found numerically by run B); Question 1.4 for n ≤ 2 is
  proved in Section 5 (the n = 2 formula had been checked by both runs).
- These additions were not examined by the two runs; `check_note.py` checks their identities.

## Third independent verification run (on the final text)
A third independent verification run, also AI-assisted, examined the written note, this package and the sources on
16 October 2026, as if nothing had been checked before; its programs and outputs are in
`reproducibility/independent_run_2/` (with a README).
- **Sources read again.** Problem 4.23, Update 4.23 and Tables 1 and 2 of arXiv:1809.07200v2 (local copy, sha256
  8e28fd44…, equal to the project's manifest); erdosproblems.com #509 and #1046 (fetched again, anonymously);
  Erdős–Hwang 1978 in full; Section 5 of Jahn–Martini–Richter; Definition 0.1 and Theorems 0.2–0.3 of
  Bauer–Catanese–Di Scala; Erdős 1961 pp. 246–247; the arXiv abstract of Micu–Pintea–Ţurcaş. Pommerenke 1959 and
  Cuyt–Driver–Lubinsky 1996 were not fetched again (automated access is refused); the note's statements about them
  agree with the reading recorded when the note was written.
- **Proofs.** Every proof re-derived line by line: Theorem 1.1, Lemmas 2.1 and 2.2 (including the equality case of
  Bieberbach's theorem and the use of Rudin's Theorems 13.11 and 14.8), Remark 2.3 (the derivation of the non-strict
  form from Pommerenke's Theorem 3), Remarks 3.1–3.4, Proposition 1.2, Corollary 1.3, the case n ≤ 2 of Question 1.4,
  and Lemma 6.1; `check_note.py` read line by line (the covering of the spheres, the 1-Lipschitz radial projection,
  the cell bound, the subharmonicity step, the counts 87 = 37 + 28 + 22). No error, no gap.
- **Own programs** (all checks pass): Proposition 1.2 exactly for n ≤ 20, with a second proof of the upper bound
  (|T_n(e^{iθ})|² − 1 has no zero in (−1, 1) as a polynomial in cos θ; maximum modulus principle for 1/T_n on
  |w| > 1), and by ray casting in R^3 at 40 digits for n ≤ 12; Remark 3.3 (exact identity, a proof of the sphere
  bound without concavity, minima on the sphere for four values of a, voxel labelling: seven components, the
  component of 0 inside |x| < a/√3); Lemma 6.1 on 64,800 segments in 108 random configurations; the one-dimensional
  problem by exhaustive grids (n = 3, 4) and differential evolution (n = 3..10, two seeds): no feasible value above
  2·2^(−1/n), which is reached to within 10⁻¹²; the n = 2 example of Section 1.1, Jung's thresholds, Chebyshev's
  inequality and widths (Remark 3.4), three further norms (Remark 3.2), the pinching of the Chebyshev open
  lemniscates (Remark 2.3), and Theorem 1.1 in R^3 on 30 configurations (largest radius 1.6343).
- **Re-run of the package.** All 22 commands of `reproducibility/README.md` from an extracted copy of the source
  archive: outputs identical to the recorded ones up to running times (`independent_run_2/package_rerun/`).
- **Literature.** arXiv API (13 queries, among them the complete lists for "lemniscate" and "lemniscates", 311
  titles), zbMATH Open (8 queries), Crossref (the ten DOIs of the bibliography), OpenAlex (work lookups; the lists of
  citing works were refused, HTTP 429), Semantic Scholar (works citing Pommerenke's two 1959 papers), one web search.
  No proof of the covering statement for m ≥ 3 was found.
- **Corrections required, all made.** (1) Abstract: the origin of the question ("posed in Hayman's collection of
  1974", now also sourced to Table 2 of the arXiv edition) instead of "according to erdosproblems.com, first posed";
  (2) "best radius about the centroid" instead of "best constant", and the value 2·2^(−1/n) is open only for n ≥ 3;
  (3) the sharpness is no longer listed among the contributions of the reduction, and the Chebyshev example is not
  claimed as new for m ≤ 2; (4) Remark 3.3: the level is called κ, the radius of the sphere r; (5) Section 6, part (iii):
  what the program checks for zⁿ − 1; (6) Section 5: the tolerance 10⁻⁹ of the constraints of the SLSQP search;
  (7) this run recorded in the note and in the package; (8) the Zenodo description. Optional precisions also made:
  "and is connected" in the proof of Lemma 2.1; "with n ≥ 2" and the step from the curve to the filled set in
  Remark 2.3; Theorem 0.3(2) of Bauer–Catanese–Di Scala.
- **Verdict.** No mathematical error; every recomputed number is correct; the scope (partial answer: the covering
  question answered, the volume question open) is stated correctly.

## Relation to the literature, novelty and scope
- **Searches (16 October 2026, by the first computation and the three runs).** arXiv API (the third run screened
  all 311 titles returned for "lemniscate" and "lemniscates"), zbMATH Open, Crossref, OpenAlex (works citing
  Pommerenke 1959: 5, all planar; works citing Erdős–Herzog–Piranian 1958: 56 titles; refused to the third run
  because of rate limits), Semantic Scholar (partly refused, rate limits; works citing Pommerenke's two 1959 papers:
  9 and 8, none on R^m), erdosproblems.com (#509, #1046, their forum threads, the tags "analysis" and
  "polynomials"), nine web searches. The works citing the Hayman–Lingham book were not screened.
- **What was read.** Pommerenke 1959 (Theorems 3 and 4, p. 222); Cuyt–Driver–Lubinsky 1996 (size of lemniscates in
  C and C^n; no covering statement for the sets E); Erdős 1961 pp. 246–247; Erdős–Hwang 1978 in full (inclusion of
  such sets in R^3); Jahn–Martini–Richter 2016 Section 5 (star-shapedness and components of multifocal Cassini
  sublevel sets for gauges; no covering statement); Bauer–Catanese–Di Scala 2017 (abstract, definitions, main
  theorems; full text searched for Erdős, Pommerenke, Hayman, Piranian and covering statements by run A: none);
  Micu–Pintea–Ţurcaş 2026 (abstract only: convexity of high sublevel sets of ∏|x − p|²).
- **Not read.** Erdős–Herzog–Piranian 1958; Pommerenke, "On some metric properties of polynomials with real zeros"
  (Michigan Math. J. 6 (1959) 377–380), which Erdős 1961 lists among the references; Hayman 1974; the 2019 book;
  the CAGD papers on three-dimensional lemniscates cited by Jahn–Martini–Richter.
- **Novelty.** No printed proof of the covering statement in R^m, m ≥ 3, was found in the sources searched. The
  argument is short and classical, so a published version may exist; a search that finds nothing is not a proof of
  novelty, and no priority is claimed.
- **Scope.** The covering question of Problem 4.23 is answered; the volume question and Question 1.4 are open.

## 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/