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bibkey: mane2026vanishing
authors: S. R. Mane
year: 2026
title: Identically vanishing k-generalized Fibonacci polynomials
doi: null
url: https://arxiv.org/abs/2507.11596v4
claim: Conjecture 6.6 asserts a negative-index root-amplitude upper bound and attainment at every negative multiple of k; the attainment clause fails at k = 3 and n = -15.
strata_touched:
  - D5/S3/Zeros/NegativeIndexTribonacciAttainmentRefutation
license: citation-only
triage: anchor

Verified locator

DOI: null

Source: https://arxiv.org/abs/2507.11596v4

The inspected source is version 4. Equations (1.2)–(1.3), p. 2, give the forward and backward polynomial recurrences. Equations (1.4a)–(1.4b), p. 3, define the quotient and remainder. Conjecture 6.4 and the paragraph defining ζ occur on p. 19; Conjectures 6.5–6.6 occur on p. 20.

Crossref’s title query returned no exact matching work among its first twenty results; no Crossref DOI is assigned here.

Source definitions

The recurrence (1.2) is

$${\mathcal{F}}_{n,k}(x)=x^{k-1}{\mathcal{F}}_{n-1,k}(x)+x^{k-2}{\mathcal{F}}_{n-2,k}(x)+\cdots +{\mathcal{F}}_{n-k,k}(x).$$

“By convention, the initial values are 𝓕_{n,k}(x) = 1 for n = 1 and 𝓕_{n,k}(x) = 0 for n ∈ [−(k−2), 0].” (p. 2.)

“For k ≥ 2 and n ∈ ℤ, let ζ_{n,k} denote the maximum amplitude of |x_root|^k, where P_{n,k}(x_root^k) = 0. Set ζ_{n,k} = 0 if 𝓕_{n,k}(x) has no nonzero roots, including vanishing polynomials.” (p. 19.)

Conjecture 6.6

“For fixed k ≥ 3 and n < 0, the upper bound is ζ_{n,k} ≤ ⌊|n|/k⌋. The bound is attained whenever r_{n,k} = 1 (equivalently n = −sk, where s ≥ 1), i.e. it is a tight bound.” (p. 20.)

The encoded proposition is only the attainment clause, with all natural k ≥ 3 and s ≥ 1. The backward recurrence uses a k m = 𝓕_{1−m,k} and F k n = a k (1−n).toNat. The statistic is Real sSup of the image of the nonzero complex root set under z ↦ ‖z‖^k. The empty-set and unbounded-set totalizations both give zero, including the zero-polynomial case for k ≥ 2.

Refutation and scope

The recurrence gives

$${\mathcal{F}}_{-15,3}(x)=-x(x^{12}+8x^9+18x^6+15x^3+5).$$

Every nonzero complex root has |x|³ < 5. The amplitude set is finite and nonempty, so ζ_{−15,3} < 5. This refutes attainment at the nondegenerate instance k = 3, s = 5. It preserves the upper-bound inequality at that instance and does not settle the universal upper-bound clause or the branch slope assertions in Conjecture 6.6. Conjectures 6.4 and 6.5 are separate statements; their validity is not a premise of this refutation.