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arxiv: 2606.21628 · v1 · pith:CVI6YUUAnew · submitted 2026-06-19 · 🧮 math.NT · math.AC· math.AG

On Bounds of Extension Degrees for Similarity of Integral Matrices over Number Fields

Mingqiang Feng , Ziyang Zhu This is my paper

Pith reviewed 2026-06-26 13:08 UTC · model grok-4.3

classification 🧮 math.NT math.ACmath.AG
keywords integral matricesmatrix similaritynumber fieldsextension degreeslocal-global principlecharacteristic polynomialseparable polynomialsrings of integers
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The pith

No uniform bound exists on the extension degree needed to make locally similar integral matrices similar over the integers of a number field extension.

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

The paper establishes that if two n by n integral matrices over a number field K are similar over every completion of its ring of integers, then they become similar over the ring of integers of some finite extension of K. It proves this extension degree cannot be bounded uniformly across all such matrices. When the characteristic polynomial is fixed and separable, however, an explicit upper bound on the degree does exist. A reader cares because this distinguishes the general local-to-global principle for matrix similarity from its effective, bounded version under extra invariants.

Core claim

While local similarity over all completions implies global similarity after a finite extension of K, there is no uniform bound on the degree of that extension valid for all n by n matrices; an upper bound does exist when the characteristic polynomial is given and separable.

What carries the argument

The finite extension of K making local similarities imply global similarity over the ring of integers, with its degree controlled or uncontrolled by the characteristic polynomial.

If this is right

  • For matrices without a fixed characteristic polynomial, the required extension degree can be made arbitrarily large by choosing suitable examples.
  • When the characteristic polynomial is fixed and separable, the extension degree is bounded above by a quantity depending only on that polynomial and n.
  • Similarity of integral matrices can be decided by checking local conditions and then passing to a controlled extension.
  • The result separates the existence of some extension from the existence of a uniform or polynomial-dependent bound.

Where Pith is reading between the lines

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

  • The bound for separable characteristic polynomials might allow algorithmic checks of matrix similarity by enumerating extensions up to that degree.
  • The lack of a uniform bound suggests that effective local-global principles for GL_n or related groups over number fields require additional invariants beyond dimension.
  • One could test whether the bound remains finite when the characteristic polynomial is allowed to vary within a fixed degree or with bounded coefficients.

Load-bearing premise

Local similarity over all completions of the ring of integers always implies global similarity after some finite extension.

What would settle it

A sequence of pairs of n by n integral matrices over varying number fields where the smallest extension degree making them similar grows without bound as the pairs vary.

read the original abstract

It is well-known that if $n\times n$ integral matrices $A$ and $B$ of a number field $K$ are similar over all completions of the ring of integers of $K$, then $A$ and $B$ are similar over the ring of integers of a finite extension of $K$. We prove that there is no uniform bound of the degree of extension of $K$ valid for all $n\times n$ matrices. On the other hand, we provide a upper bound of the degree of extension of $K$ for a given separable characteristic polynomial.

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

0 major / 3 minor

Summary. The paper recalls the standard local-to-global principle that n×n integral matrices A, B over a number field K which are similar over every completion of the ring of integers of K become similar over the ring of integers of some finite extension L/K. It proves there is no uniform bound on [L:K] that works for all n×n matrices, but supplies an explicit upper bound on [L:K] when the common characteristic polynomial is fixed and separable.

Significance. The results clarify the quantitative behavior of the local-to-global principle for integral matrix similarity. The absence of a uniform bound shows that the extension degree can be arbitrarily large in general, while the bound for fixed separable characteristic polynomials gives a concrete control that may be useful in arithmetic applications. The argument rests on a well-known fact treated as given and derives the two main statements from it without apparent internal inconsistency.

minor comments (3)
  1. The abstract states the two main results clearly but the manuscript should include a brief statement of the well-known local-global fact (with reference) at the beginning of the introduction or §1 to make the logical structure self-contained.
  2. Notation for the ring of integers and its completions should be fixed consistently throughout; the abstract uses “the ring of integers of K” while later sections may introduce O_K or similar, which can be clarified in a notation paragraph.
  3. The bound for the separable case is described as “an upper bound”; if an explicit expression or dependence on n and the polynomial is derived, it should be stated in the abstract and highlighted in the introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for the recommendation of minor revision. The referee's summary correctly reflects the paper's content: the local-to-global principle for similarity of integral matrices, the non-existence of a uniform bound on the extension degree, and the explicit bound provided when the characteristic polynomial is fixed and separable. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper's central results rest on the standard local-to-global principle for matrix similarity over number fields, which is explicitly labeled 'well-known' and treated as an external given rather than derived or cited from the authors' prior work. The no-uniform-bound result and the separable-polynomial bound are then derived from this premise without any self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations. No equations or steps reduce to the paper's own inputs by construction. This is the normal case of an independent argument built atop established external facts.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard local-global implication for matrix similarity and on the definition of separability for the characteristic polynomial.

axioms (1)
  • domain assumption If two integral matrices are similar over all completions of the ring of integers of K, then they are similar over the ring of integers of some finite extension of K.
    Explicitly labeled 'well-known' in the abstract and used as the starting point for both the negative and positive results.

pith-pipeline@v0.9.1-grok · 5622 in / 1229 out tokens · 23573 ms · 2026-06-26T13:08:16.640640+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

8 extracted references · 1 canonical work pages · 1 internal anchor

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