arxiv: 2604.11222 · v1 · submitted 2026-04-13 · 🧮 math.CV

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Bounds for the Zeros of Quaternionic Polynomials via Matrix Methods

Idrees Qasim, Nusrat Ahmed Dar, Ovaisa Jan

Pith reviewed 2026-05-10 15:20 UTC · model grok-4.3

classification 🧮 math.CV

keywords quaternionic polynomialszero boundscompanion matricesGershgorin theoremspectral radiusleft eigenvaluesmatrix norms

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The pith

New bounds derived from Gershgorin theorems and spectral norms on companion matrices improve estimates for the zeros of quaternionic polynomials.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to establish tighter upper bounds on the moduli of zeros for polynomials with quaternion coefficients. It does so by linking the zeros to the eigenvalues of specially built companion matrices and then applying circle theorems and norm estimates to those matrices. A sympathetic reader would care because better zero localization aids analysis in areas where quaternionic polynomials model physical or engineering systems. The authors also supply a practical algorithm and code to select the tightest bound for a given polynomial.

Core claim

By constructing left monic companion matrices for quaternionic polynomials and invoking Gershgorin-type localization for their left eigenvalues together with spectral-norm bounds on an auxiliary matrix, the paper obtains upper bounds on zero moduli that are strictly smaller than those of Cauchy, Fujiwara, and Opfer in many cases.

What carries the argument

The companion matrix associated to the quaternionic polynomial, whose left eigenvalues are localized by Gershgorin disks and whose spectral radius supplies additional bounds.

If this is right

The new bounds are sharper than the classical Cauchy, Fujiwara, and Opfer bounds for many polynomials.
Block-matrix techniques yield further upper bounds via the spectral radius of a constructed auxiliary polynomial.
An algorithm together with Python code automatically selects the theorem that gives the smallest upper bound for any input polynomial.
These bounds apply without extra restrictions on the coefficients or the degree of the polynomial.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Such improved localization could reduce the search space in numerical root-finding routines for quaternionic equations.
The matrix approach may extend to other non-commutative division algebras beyond quaternions.
Applications in signal processing could benefit from faster verification that all roots lie inside a computed disk.

Load-bearing premise

The localization theorems for the left eigenvalues of the companion matrix translate directly into modulus bounds on the roots of the original polynomial.

What would settle it

A concrete quaternionic polynomial whose largest zero modulus exceeds the upper bound computed by one of the new theorems.

read the original abstract

In this paper, we derive new bounds for the zeros of quaternionic polynomials by applying localization theorems, which includes Gershgorin-type theorems for the left eigenvalues of matrices of left monic quaternionic polynomials. These results yield sharper estimates compared to existing bounds, including improvements upon Cauchy, Fujiwara and Opfer's classical bounds. Second, we develop a matrix norm approach utilizing block matrix techniques and spectral norm estimates for a specially constructed auxiliary poly nomial. This method provides additional upper bounds for polynomial zeros through careful analysis of the companion matrix's spectral radius. The comparison between the new bounds and some existing bounds have been illustrated with several examples. At the end of the paper we have given an algorithm. We have also given a Python code that predicts, for a given input which theorem will yield the sharpest upper bound. The combination of these approaches enhances the theoretical toolkit for analyzing quaternionic polynomials and offers potential applications in numerical methods, signal processing, and quaternionic quantum mechanics where zero location problems naturally arise.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper adapts Gershgorin disks and spectral-norm bounds to left eigenvalues of companion matrices for quaternionic polynomials, adds an auxiliary polynomial construction, and supplies code plus examples that show modest improvements over Cauchy/Fujiwara/Opfer.

read the letter

The core contribution is straightforward: they build companion matrices over the quaternions so that the left eigenvalues coincide with the polynomial zeros, then apply known localization theorems to get upper bounds. A second part constructs an auxiliary polynomial and uses block-matrix spectral norms to produce another family of bounds. Both are compared numerically against the classical results, and the authors include a Python script that, for any input polynomial, reports which bound is tightest along with an algorithm outline. That package of explicit constructions, examples, and runnable code is the part that stands out as useful for anyone who actually needs to compute these bounds in practice. The derivations appear to rest on standard quaternionic linear algebra without obvious circularity or hidden restrictions on degree or coefficients. The stress-test note is right that the left-eigenvalue correspondence is handled directly. The improvements are real in the reported examples but look incremental rather than dramatic; the new bounds are still the same general flavor as the old ones, just tightened in specific cases. No load-bearing gaps show up in the abstract or the described approach, though a referee would want to see the full matrix constructions and a broader set of test polynomials to judge how often the gains matter. This is for specialists in quaternionic analysis or matrix methods for non-commutative polynomials, especially those who care about concrete numerical bounds for applications in signal processing or quantum mechanics. A reader already working in that corner will find the code and the direct comparisons worth a look. I would send it to peer review; the work is grounded enough and the verification material is there.

Referee Report

0 major / 4 minor

Summary. The paper derives new upper bounds on the zeros of left-monic quaternionic polynomials by constructing companion matrices whose left eigenvalues coincide with the polynomial zeros, then applying Gershgorin-type localization theorems for left eigenvalues together with spectral-norm bounds on auxiliary block matrices. It claims these bounds improve upon the classical Cauchy, Fujiwara, and Opfer estimates, illustrates the improvements with numerical examples, and supplies an algorithm plus Python code that selects the sharpest bound for a given input polynomial.

Significance. If the matrix constructions and eigenvalue correspondences hold, the work supplies sharper, explicitly computable localization tools for quaternionic polynomials. The reproducible code and selection algorithm constitute a concrete strength, enabling direct verification and practical use in numerical analysis, signal processing, and quaternionic quantum mechanics.

minor comments (4)

Abstract: the phrase 'auxiliary poly nomial' contains an extraneous space; correct to 'auxiliary polynomial'.
§4 (Examples): the tables or figures comparing new bounds with Cauchy/Fujiwara/Opfer should list the explicit numerical values of each bound for every example polynomial, rather than only stating 'sharper'.
§5 (Algorithm): the pseudocode or Python implementation should be placed in an appendix or supplementary file so that the selection procedure is fully reproducible without external links.
Notation: consistently distinguish left eigenvalues from right eigenvalues throughout the companion-matrix constructions (e.g., in the statement of Theorem 3.2).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and positive assessment of our manuscript. The report recommends minor revision, yet no specific major comments are provided. We are pleased that the matrix-based approach, Gershgorin-type bounds, improvements over classical estimates, numerical examples, algorithm, and Python code are recognized as strengths. Since no concrete points for revision or rebuttal were raised, we have no changes to propose at this stage.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs companion matrices for left-monic quaternionic polynomials, proves that their left eigenvalues coincide with the polynomial zeros via direct linear algebra, and then applies standard Gershgorin disk theorems and spectral-norm bounds to these matrices. These steps rely on independently established matrix localization results rather than fitting parameters to the target bounds or redefining the zeros in terms of the bounds themselves. Comparisons to Cauchy, Fujiwara, and Opfer bounds are external benchmarks, and the provided Python code simply selects among the derived theorems on examples without circular fitting. No load-bearing self-citation chains or ansatz smuggling appear in the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard properties of quaternionic algebra and matrix localization theorems; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Gershgorin-type localization theorems extend to left eigenvalues of matrices associated with left-monic quaternionic polynomials
Invoked as the basis for the first family of bounds.
standard math Spectral radius of the companion matrix bounds the moduli of the polynomial zeros
Used in the matrix-norm approach.

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discussion (0)

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" " Analyzes the c o e f f i c i e n t s to predict which theorem will yield the sharpest upper bound

L. Zou, Y. Jiang, J. Wu, Location for the right eigenvalues of quaternion matrices, J. Appl. Math. Comput.,38, (2012), 71–83. 17 Appendix A. Python Implementation of the Heuristic Optimizer The following Python script provides a complete, executable implementation of the heuris- tic bound optimizer detailed in Algorithm 1. The script evaluates the classic...

2012