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arxiv: 2604.11215 · v1 · submitted 2026-04-13 · 🧮 math.CV

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Bounds for the Zeros of Polynomials over Quaternion Division Algebra

Idrees Qasim, Ovaisa Jan

Pith reviewed 2026-05-10 15:24 UTC · model grok-4.3

classification 🧮 math.CV
keywords quaternionic polynomialszero boundsspectral normcompanion matricespartitioned matricesquaternion division algebrazero localization
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The pith

New spectral norm inequalities for partitioned quaternionic matrices yield sharper upper bounds on 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 locate the zeros of polynomials whose coefficients are quaternions more accurately than before. It first proves fresh inequalities that bound the spectral norms of quaternionic matrices when they are arranged in partitioned blocks. These inequalities are then applied to the companion matrices of the given polynomials, including powers of those matrices, to produce explicit upper bounds on the size of any zero. A reader would care because quaternionic polynomials arise in modeling rotations and orientations, and tighter zero bounds restrict the regions where numerical solvers must search. The authors state that the resulting bounds improve on earlier estimates and work uniformly for polynomials of any degree.

Core claim

By establishing novel spectral norm inequalities for partitioned quaternionic matrices and utilizing the structural properties of companion matrices and their higher powers, we derive unexplored upper bounds for the zeros of quaternionic polynomials. Our bounds are systematically sharper than existing results and provide a unified framework for zero localization in the quaternionic setting.

What carries the argument

Spectral norm inequalities for partitioned quaternionic matrices, which are applied to companion matrices to bound the moduli of polynomial zeros via control of matrix power growth.

Load-bearing premise

The newly proved spectral norm inequalities for partitioned quaternionic matrices continue to hold for the exact block structures and non-commutative multiplication rules that appear inside the companion matrices.

What would settle it

Construct any explicit quaternionic polynomial, compute one of its actual zeros by solving the equation, and check whether its modulus exceeds the upper bound given by the new inequalities; a single counter-example would refute the claim.

read the original abstract

Locating the zeros of quaternionic polynomials is a fundamental problem with significant implications across scientific and engineering disciplines, yet the noncommutative nature of quaternion multiplication makes it fundamentally more complex than the classical complex case. In this paper, we develop new bounds for the zeros of polynomials with quaternionic coefficients. We establish spectral norm inequalities for quaternionic matrices, particularly those of a partitioned form. These inequalities are applied to specialized quaternionic companion matrices to derive novel upper bounds for the zeros of the original polynomial. By establishing novel spectral norm inequalities for partitioned quaternionic matrices and utilizing the structural properties of companion matrices and their higher powers, we derive unexplored upper bounds for the zeros of quaternionic polynomials. Our bounds are systematically sharper than existing results and provide a unified framework for zero localization in the quaternionic setting.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to derive novel upper bounds for the zeros of polynomials with quaternionic coefficients. It first establishes spectral norm inequalities for partitioned quaternionic matrices, then applies these inequalities to specialized quaternionic companion matrices and their higher powers to obtain the zero bounds. The abstract asserts that the resulting bounds are systematically sharper than existing results and provide a unified framework for zero localization in the quaternionic setting.

Significance. If the central claims hold after verification, the work would extend classical matrix-norm techniques for polynomial zero bounds to the non-commutative quaternionic case, which could be useful in applications involving quaternionic polynomials. The approach of using companion matrices is standard, but the novelty would lie in the partitioned spectral-norm inequalities and their applicability; however, the lack of explicit comparisons or proofs in the abstract limits assessment of the advance.

major comments (2)
  1. [Abstract] Abstract: The claim that the bounds are systematically sharper than existing results is not supported by any explicit comparisons, numerical data, or analytical arguments within the provided abstract, making it impossible to evaluate the improvement over prior quaternionic or complex bounds.
  2. [Main Results (application to companion matrices)] The central derivation applies the new spectral norm inequalities for partitioned quaternionic matrices to companion matrices, but the manuscript does not verify that the rigid block-shift structure of the companion matrix (with identity blocks on the subdiagonal and coefficient blocks in the final column) satisfies the partitioning hypotheses used to derive the inequalities, particularly under non-commutative multiplication.
minor comments (1)
  1. [Abstract] The abstract repeats the main claim twice with nearly identical wording; this should be consolidated for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below and indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that the bounds are systematically sharper than existing results is not supported by any explicit comparisons, numerical data, or analytical arguments within the provided abstract, making it impossible to evaluate the improvement over prior quaternionic or complex bounds.

    Authors: We agree that the abstract as written asserts the improvement without direct supporting material. In the revised manuscript we will modify the abstract to reference the explicit comparisons and numerical examples already developed in Section 4 (and the analytical arguments in Section 3), or, if space is limited, we will qualify the statement to indicate that sharpness is demonstrated through the derivations and concrete examples in the body of the paper. revision: yes

  2. Referee: [Main Results (application to companion matrices)] The central derivation applies the new spectral norm inequalities for partitioned quaternionic matrices to companion matrices, but the manuscript does not verify that the rigid block-shift structure of the companion matrix (with identity blocks on the subdiagonal and coefficient blocks in the final column) satisfies the partitioning hypotheses used to derive the inequalities, particularly under non-commutative multiplication.

    Authors: We thank the referee for highlighting this verification gap. We will insert a short dedicated paragraph immediately before the application of the inequalities (in the revised Section 3) that explicitly checks the block-shift structure of the quaternionic companion matrix against the partitioning hypotheses. The verification will confirm that the identity blocks on the subdiagonal and the coefficient blocks in the final column meet the required row/column partitioning conditions, and we will note that the spectral-norm inequalities remain valid under the standard non-commutative quaternion multiplication because the norm is submultiplicative and the partitioning is defined entrywise with respect to the quaternion algebra. revision: yes

Circularity Check

0 steps flagged

No circularity: independent matrix-norm inequalities applied to companion matrices

full rationale

The derivation begins by proving new spectral norm inequalities for partitioned quaternionic matrices, then applies those inequalities to the rigid block structure of companion matrices (and their powers) to bound polynomial zeros. No step defines a quantity in terms of the target bound, fits a parameter to the zeros and renames it a prediction, or relies on a self-citation chain for the load-bearing inequalities or their applicability. The chain is a standard forward proof from matrix analysis to root localization and remains self-contained against external verification of the norm bounds.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract alone supplies no explicit free parameters, ad-hoc constants, or newly invented entities; the work appears to rest on standard facts about quaternion algebra and matrix norms.

axioms (1)
  • domain assumption Spectral norms and block-matrix inequalities behave as in the commutative case once the appropriate left/right multiplication conventions are fixed.
    Implicit in the application of norm inequalities to companion matrices.

pith-pipeline@v0.9.0 · 5435 in / 1293 out tokens · 35391 ms · 2026-05-10T15:24:50.087485+00:00 · methodology

discussion (0)

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Reference graph

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