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arxiv: 2604.20929 · v1 · submitted 2026-04-22 · 🧮 math.AC

Recognition: unknown

Smith Form Equivalence for Several Classes of Multivariate Polynomial Matrices

Dongmei Li, Jiancheng Guan, Zuo Chen

Pith reviewed 2026-05-09 23:07 UTC · model grok-4.3

classification 🧮 math.AC
keywords multivariate polynomial matricesSmith formalgebra isomorphismsQuillen-Suslin theoremLin-Bose theoremGröbner basismatrix equivalence
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The pith

Algebra isomorphisms provide criteria for reducing classes of multivariate polynomial matrices to Smith form.

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

The paper investigates when several classes of multivariate polynomial matrices are equivalent to their Smith forms. It uses isomorphisms of algebras as the primary tool to establish criteria for this equivalence. The results are then extended to non-square matrices and rank-deficient matrices by applying the Quillen-Suslin and Lin-Bose theorems. These criteria can be verified algorithmically using Gröbner basis computations.

Core claim

By employing algebra isomorphisms, criteria are established for the equivalence of certain multivariate polynomial matrices to their Smith forms, with extensions to non-square and rank-deficient cases achieved through the Quillen-Suslin theorem and the Lin-Bose theorem, all verifiable via existing Gröbner basis methods.

What carries the argument

Algebra isomorphisms between the relevant rings, used to study and characterize the Smith form equivalence relation for the polynomial matrices.

Load-bearing premise

That the algebra isomorphisms between the rings preserve the Smith form equivalence for the considered matrix classes without requiring extra conditions on the matrix coefficients or degrees.

What would settle it

Identification of a multivariate polynomial matrix from one of the classes that satisfies the isomorphism condition but is not equivalent to its Smith form under elementary row and column operations.

read the original abstract

This paper investigates the equivalence reduction for several classes of multivariate polynomial matrices and their Smith forms, establishing some criteria for such reduction. In particular, we employ algebra isomorphisms as a key tool to study this equivalence problem. We then leverage the Quillen-Suslin and Lin-Bose theorems to extend these results to non-square and rank-deficient matrices. Moreover, the verification of our criteria is achievable algorithmically via existing Gr\"{o}bner basis methods.

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 / 2 minor

Summary. The paper claims to establish criteria for equivalence to Smith form for several (unspecified in the abstract) classes of multivariate polynomial matrices by using algebra isomorphisms to reduce the problem, then extending the results to non-square and rank-deficient matrices via the Quillen-Suslin and Lin-Bose theorems; it further asserts that the criteria are algorithmically verifiable using Gröbner basis methods.

Significance. If the isomorphisms are shown to preserve unimodular equivalence classes and the Smith diagonal property, and if the extensions via Quillen-Suslin and Lin-Bose are rigorously justified for the chosen classes, the work could provide a useful reduction technique in commutative algebra and systems theory. The algorithmic verification claim is a potential strength, but the manuscript supplies no concrete examples, counterexamples, or explicit commutation proofs in the abstract, limiting immediate impact.

major comments (2)
  1. [Main results section (isomorphism application)] The central reduction via algebra isomorphisms (described in the abstract as the key tool) requires explicit verification that the isomorphism maps left/right unimodular equivalence classes to equivalent classes over the target ring while preserving the existence of a diagonal Smith form. Without such a check (e.g., on degree or coefficient compatibility), the criteria may fail to transfer for arbitrary elements of the chosen classes.
  2. [Extension via Lin-Bose theorem] The extension to rank-deficient matrices via the Lin-Bose theorem is stated to follow from the isomorphism reduction, but the manuscript must demonstrate that the isomorphism preserves rank and the relevant module properties; otherwise the application is not load-bearing for the non-square case.
minor comments (2)
  1. [Abstract and Introduction] The abstract refers to 'several classes' without naming them; the introduction should list the precise classes (e.g., by degree bounds or coefficient field restrictions) at the outset.
  2. [Notation and Preliminaries] Notation for the polynomial ring (e.g., k[x1,...,xn]) and matrix dimensions should be fixed early and used consistently when stating the isomorphism and the Smith form.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive comments on our manuscript. We address each of the major comments in detail below and outline the revisions we plan to make.

read point-by-point responses

  1. Referee: [Main results section (isomorphism application)] The central reduction via algebra isomorphisms (described in the abstract as the key tool) requires explicit verification that the isomorphism maps left/right unimodular equivalence classes to equivalent classes over the target ring while preserving the existence of a diagonal Smith form. Without such a check (e.g., on degree or coefficient compatibility), the criteria may fail to transfer for arbitrary elements of the chosen classes.

    Authors: We acknowledge the importance of explicitly verifying that the algebra isomorphisms preserve the relevant equivalence classes and the Smith form property. In the revised version, we will expand the main results section to include a detailed proof or argument showing that the isomorphisms map left and right unimodular matrices to unimodular matrices over the target ring and that the Smith diagonal form is preserved. This will include considerations of degree and coefficient compatibility for the classes of matrices under study. revision: yes

  2. Referee: [Extension via Lin-Bose theorem] The extension to rank-deficient matrices via the Lin-Bose theorem is stated to follow from the isomorphism reduction, but the manuscript must demonstrate that the isomorphism preserves rank and the relevant module properties; otherwise the application is not load-bearing for the non-square case.

    Authors: We agree that a demonstration of rank preservation and module properties under the isomorphism is essential for the extension to be rigorous, particularly for the non-square and rank-deficient cases. We will add this justification in the section discussing the application of the Lin-Bose theorem, showing how the isomorphism maintains the necessary properties to allow the extension via Quillen-Suslin and Lin-Bose theorems. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external theorems

full rationale

The paper establishes criteria for Smith form equivalence of certain multivariate polynomial matrix classes by employing algebra isomorphisms as a reduction tool, then extends to non-square and rank-deficient cases via the Quillen-Suslin theorem (projective modules over polynomial rings are free) and Lin-Bose theorem. These are independent, well-known external results from algebraic K-theory and systems theory, not self-citations or internal definitions. The verification is algorithmic via Gröbner bases, which are standard independent methods. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim depends on the applicability of algebra isomorphisms to the chosen matrix classes and on the two cited theorems holding in the rings under study; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Quillen-Suslin theorem applies to the polynomial rings considered
    Invoked to extend results to non-square matrices.
  • standard math Lin-Bose theorem applies to rank-deficient matrices
    Invoked to extend results to rank-deficient cases.

pith-pipeline@v0.9.0 · 5362 in / 1262 out tokens · 29849 ms · 2026-05-09T23:07:39.933154+00:00 · methodology

discussion (0)

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Reference graph

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