bibkey: altizio2025stern
authors: David Altizio
year: 2025
title: Zeros of Stern polynomials in the complex plane
doi: 10.48550/arXiv.2511.03847
url: https://arxiv.org/abs/2511.03847v1
claim: Remark 4 asks whether the bound |(z)_n| ≥ n|z| on |z − 2| ≤ 1 holds exactly for n = 5, 6, 7.
strata_touched:
  - D5/S3/Zeros/AltizioGeometricSumDisk
license: citation-only
triage: anchor

Geometric sums on the Stern exclusion disk

Verified locator

DOI: https://doi.org/10.48550/arXiv.2511.03847

Source: https://arxiv.org/abs/2511.03847v1

Source statement

Section 2, p. 4: “For all positive integers n, we introduce the (non-standard) notation (λ)_n := 1 + λ + ⋯ + λ^{n−1} = (λ^n − 1)/(λ − 1) ∈ ℤ[λ].” The source sets (1)_n = n by continuity.

Section 3: “B := B_1[2] = {|w − 2| ≤ 1}”.

Section 4.3, p. 20, Remark 4: “One might suspect that |(z)n| ≥ n|z| more generally whenever z ∈ B. However, numerical evidence suggests that the value min{z∈B} |(z)_n/z| approaches 2π as n → ∞. It seems that this inequality is true only for n = 5, 6, and 7, but we do not have a proof.”

Mathematical scope

The finite sum is ∑ k ∈ Finset.range n, z ^ k for positive natural n. The disk is Metric.closedBall (2 : ℂ) 1. The classification quantifies over every point of this closed disk and both directions of the degree criterion. Proposition 8 supplies the source’s degree-five and degree-six bounds. The repository result proves the classification, including degree seven and the excluded positive degrees. The 2π limit and the separate inequality in Remark 3 are outside that result.

Supporting computation and literature boundary

The exact-arithmetic experiment is in the-omega-institute/trureturing-experiments, commit 65ee1fc450d3e106194a87a31f910d2ef1ba35bd, at docs/reports/altizio-2025-geometric-sum-disk/check.py, SHA-256 cc441905676ba7ed3924ce669b140bf6869da609f92bfcfe8b162a57cd26181d. It exits 0 and checks the three polynomial certificates, the source’s Proposition 8 identities, and the excluded-degree coefficient and rational parameters through degree 60. These finite checks do not prove the unbounded classification.

The literature screen on issue #20656 includes the arXiv version and the author’s dissertation. Its completeness is ASSUMED-UNVERIFIED; the source is cited as posing the question, not as proving its answer.