Simple polynomial equations over $(2 \times 2)$-matrices

Alexandre Karassev; Vitalij A. Chatyrko

REVIEW 2 major objections 5 minor 1 cited by

Simple polynomial equations over $(2 \times 2)$-matrices

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For a polynomial equation with scalar coefficients over real 2x2 matrices, every solution is similar to a diagonal, Jordan, or rotation-scaled matrix, and the full solution set has dimension exactly 2.

desk verdict A correct, small paper on solution sets of scalar polynomial equations over real 2x2 matrices; the dimension theorem is right, but Corollary 4.4 needs an added argument for the non-closed Jordan class. read the letter →

arxiv 2506.07689 v1 pith:6YDNENJ2 submitted 2025-06-09 math.RA math.GN

Vitalij A. Chatyrko , Alexandre Karassev This is my paper

classification math.RAmath.GN MSC 15A2454F4515B30

keywords polynomialmatrixequations2x2matricesrealJordannormalformsimilarityclassescoveringdimensionsolutionsetalgebraconjugacy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a complete classification of the solution set of a polynomial equation $X^n + a_{n-1}X^{n-1}+\cdots+a_0 I = O$ in real $2\times 2$ matrices, where the coefficients are real scalars and $I$ is the identity. The solutions are exactly the similarity classes of three normal forms determined by the roots of the scalar polynomial $f_s(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$: diagonal matrices for pairs of real roots, Jordan blocks for repeated real roots, and rotation-scaled blocks for complex conjugate roots. The paper also shows the solution set, as a subset of $\mathbb{R}^4$, has covering dimension $2$. This matters because it reduces a potentially complicated nonlinear question about matrices to finite bookkeeping of root patterns, and settles the dimension question left open for this scalar-coefficient case.

What carries the argument

The load-bearing mechanism is the correspondence between the scalar polynomial $f_s$ and the three real canonical forms from the real Jordan decomposition: diagonal matrices $D(p,q)=\mathrm{diag}(p,q)$, Jordan blocks $J(p)$ with eigenvalue $p$ and a single off-diagonal $1$, and rotation-scaled blocks $R(a,b)$ with first row $a,b$ and second row $-b,a$. Lemma 2.3 evaluates $F_s$ on these forms directly, turning the matrix equation into the scalar equation $f_s$ at the entries, and Lemma 2.4 uses conjugation invariance to carry solutions to whole orbits $S(B)$. For the dimension count, the quadratic case $X^2+a_1X+a_0I=O$ is solved explicitly in terms of a parameter region in $\mathbb{R}^2$, showing each non-scalar orbit is two-dimensional; the countable sum theorem then assembles the union.

What would settle it

Take $f_s(x)=x^2-1$ and compute the covering dimension of the conjugacy class $S(D(-1,1))$ directly from its definition as the image of the group of invertible $2\times 2$ matrices under conjugation. Theorems 3.4 and 4.5 imply this dimension is 2; an independent computation yielding 1 or 3 would refute the dimension claim.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 3.4: the solution set $S$ of $F_s(X)=O$ is the disjoint union of $S(D(p,p))$ for every real root $p$ of $f_s$, $S(J(p))$ for every real root of multiplicity at least two, $S(D(p,q))$ for every pair of distinct real roots $p<q$, and $S(R(a,b))$ for every complex root $a+bi$ with $b>0$, where $S(B)$ denotes the conjugacy class of $B$, i.e. the set of all matrices $C^{-1}BC$ with $C\in GL_2(\mathbb{R})$. Theorem 4.5 adds that $\dim S=2$. So for any degree $n\ge 2$, the solution set is a finite union of two-dimensional similarity classes together with isolated points, and its covering dimension is exactly $2$.

Load-bearing premise

The dimension conclusion assumes that each non-scalar similarity class $S(D(p,q))$, $S(J(p))$, and $S(R(a,b))$ is genuinely two-dimensional; the proof establishes this only indirectly by combining the two-dimensionality of the whole solution set with the zero-dimensionality of the point classes.

Editorial extensions

If this is right

For every $n\ge 2$, the solution set of $F_s(X)=O$ is $\sigma$-compact: it is a finite disjoint union of continuous images of $\mathrm{GL}_2(\mathbb{R})$ under conjugation.
The equation $X^n=O$ has the same solution set for every $n\ge 2$, namely $S(D(0,0))\cup S(J(0))$, so nilpotent scalar equations always produce a two-dimensional set.
Powers of equations can enlarge the solution set only by adding Jordan-block classes; for example $(X^2-I)^2=O$ contains $S(J(-1))$ and $S(J(1))$ in addition to the solutions of $X^2-I=O$.
Every non-scalar similarity class $S(D(p,q))$, $S(J(p))$, and $S(R(a,b))$ contains a subset homeomorphic to $\mathbb{R}^2$, hence has the cardinality of the continuum.
When the right-hand side is a non-scalar matrix $A\ne O$ with $\det A=0$, the equation $X^2+a_0I=A$ has exactly 0, 2, or 4 solutions, a finite behavior sharply different from the scalar-coefficient case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The authors leave implicit that the same root-pattern decomposition should carry over to $m\times m$ real matrices: a scalar polynomial equation should have a solution set consisting of finitely many conjugacy classes of Jordan and real canonical forms, with dimension $2m$ whenever a non-scalar class occurs; the paper only proves this for $m=2$.
Because the classification depends only on the root multiset of $f_s$, small coefficient changes that preserve root multiplicities should leave the solution set homeomorphic, a stability property the paper does not state.
One direct test of the pattern in higher dimension is to solve $X^2+I=O$ in real $3\times 3$ matrices; extrapolating from this paper predicts a 6-dimensional solution set, computable by the same entry-wise systems used in Lemma 4.1.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

This is a small, honest paper that does exactly what it says. For F_s(X)=O with scalar coefficients, it describes the solution set as a disjoint union of similarity classes corresponding to real roots, repeated real roots, pairs of real roots, and complex-conjugate roots, and then proves dim S = 2. The classification itself is a corollary of the real Jordan form plus similarity invariance—not deeply new, but cleanly organized and usable as a reference. The genuinely new part is the dimension statement, and that is the right scale of result for a short note.

Lemma 4.1 is the workhorse. It solves the quadratic case by explicit parametrization, and the parametrizations are correct. The applications of the countable sum theorem there are legitimate because the pieces are closed. Theorem 4.5 then follows from Theorem 3.4 and Corollary 4.4.

The soft spot is Corollary 4.4, exactly where the stress-test note points. For S(D(p,q)) and S(R(a,b)) the similarity classes are closed, so “dim S=2 and the point classes have dimension 0” forces the class to have dimension 2 via the finite sum theorem. For S(J(p)) that inference does not work, because S(J(p)) is not closed: for p=0 the zero matrix lies in its closure. The paper asserts in the same corollary that these classes contain subsets homeomorphic to R^2; that assertion is true and would repair the argument, but it is left unproved. An explicit embedding for p=0, for instance phi(x,t)=[[x,t],[-x^2/t,-x]] for t>0, would close the gap. So the claimed theorem is correct, but one proof line is currently doing work it should not.

The examples in Sections 5 and 6 are consistent and useful. The paper cites standard sources; there is no self-citation inflation and no circularity. The closing questions are natural.

Who should read it: anyone wanting a citable description of solution sets for scalar polynomial equations over 2x2 real matrices, and anyone teaching dimension theory in matrix spaces. It is not a breakthrough, but it is a competent, complete solution of a natural small problem.

Recommendation: send it to a serious referee. With the Corollary 4.4 gap filled—one paragraph would do—it should be accepted. As a referee, I would flag that one point and otherwise sign off.

Referee Report

2 major / 5 minor

Summary. The paper studies the solution set S ⊂ M_2(R) of monic polynomial equations X^n + a_{n-1}X^{n-1} + ... + a_0 I = O with real scalar coefficients. Theorem 3.4 decomposes S as a disjoint union of similarity classes S(D(p,p)), S(J(p)), S(D(p,q)), and S(R(a,b)) attached respectively to real roots, real roots of multiplicity at least two, pairs of distinct real roots, and conjugate pairs of non-real roots. Lemma 4.1 and Theorem 4.3 give explicit parametrizations for the quadratic case, and the paper concludes in Theorem 4.5 that dim S = 2.

Significance. The paper gives a complete, clean classification of the solution set and settles the dimension question for all degrees. The approach is elementary and largely self-contained: it uses real Jordan normal form, similarity invariance, and an explicit parametrization of the quadratic equation. The dimensional result for such solution sets is a nice contribution to the topology of matrix equations. The proofs contain no fitted parameters and no circular reasoning. However, one key proof step in Corollary 4.4 is incomplete, and the deduction of Theorem 4.5 needs an additional covering argument, so the paper requires revision.

major comments (2)

[4, Corollary 4.4]
[4, Theorem 4.5]

minor comments (5)

[4, Corollary 4.4]
[2, Proposition 2.2]
[3, before Lemma 3.1]
[4, Lemma 4.1]
[1, Proposition 1.2]

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the solution-set description and dimension theorem are derived from standard Jordan-form facts and explicit coordinate computations, not from fitted inputs or self-citations.

full rationale

The paper contains no fitted parameters, no prediction derived from fitted data, and no load-bearing self-citations. The structural description of the solution set (Theorem 3.4) is proved from the standard real Jordan form (Proposition 2.1, cited to [L]) together with the explicit evaluations in Lemma 2.3 and the similarity invariance in Lemma 2.4; none of these cited results already contains the theorem being proved. The dimension claim is established by a direct coordinate solution of the quadratic matrix equation in Lemma 4.1, from which Theorem 4.3 obtains dimension 2 for all scalar-quadratic equations; Corollary 4.4 then transfers this to the individual conjugacy classes, and Theorem 4.5 sums the classes via Theorem 3.4 and Corollary 3.5. The only notable weakness is a proof gap, not circularity: in Corollary 4.4, the inference from dim S = 2 to dim S(J(p)) = 2 requires care because S(J(p)) is not closed, and the asserted subsets homeomorphic to R^2 are not constructed in the proof. That gap is a mathematical incompleteness, but no claim reduces to its own input by definition and no load-bearing assumption is justified solely by the authors' prior work. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted to data. The paper depends on standard linear algebra (real Jordan form), standard dimension theory, and elementary similarity invariance. No new entities such as particles, forces, or conserved quantities are introduced.

assumptions (3)

standard math Every real 2x2 matrix is similar over R to exactly one of D(p,q), J(p), or R(a,b) (real Jordan form).
Invoked as Proposition 2.1 and used as the backbone of Theorem 3.4; cited to [L] without proof in the paper.
standard math Covering dimension satisfies the countable sum theorem and the monotone theorem.
Used in Lemma 4.1 and Theorem 4.5 to control dimensions of unions and subspaces; cited to [E, Theorems 7.2.1 and 7.3.4].
standard math If matrices A and B are similar, then F_s(A)=O if and only if F_s(B)=O.
Lemma 2.4 states this invariance, which follows from polynomial arithmetic and similarity, and is used throughout to reduce arbitrary matrices to normal forms.

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Cite this review

Pith. "Pith review of Simple polynomial equations over $(2 \times 2)$-matrices." pith.science (2026). https://pith.science/paper/6YDNENJ2

@misc{pith2026250607689,
  author       = {Pith},
  title        = {Pith review of: Simple polynomial equations over $(2 \times 2)$-matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6YDNENJ2}},
  note         = {Machine review of arXiv:2506.07689}
}

abstract

We consider the polynomial equation $$X^n + a_{n-1}\cdot X^{n-1} + \dots + a_1 \cdot X + a_0 \cdot I = O,$$ over $(2 \times 2)$-matrices $X$ with the real entries, where $I$ is the identity matrix, $O$ is the null matrix, $a_i \in \mathbb R$ for each $i$ and $n \geq 2$. We discuss its solution set $S$ supplied with the natural Euclidean topology. We completely describe $S$. We also show that $\dim S =2.$

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Simple polynomial equations over (mxm)-matrices
math.RA 2025-07 conditional novelty 5.0 of 10

For real 3x3 matrices X, the set of solutions to X^n + a_{n-1}X^{n-1} + ... + a_0 I = O is a union of similarity orbits whose covering dimension is always 0, 4, or 6, or else the set is empty.

Reference graph

Works this paper leans on

4 extracted references · cited by 1 Pith paper

[1]

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R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989

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[2]

Fuchs, A

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[3]

Lutkepohl, Handbook of Matrices, John Wiley & Sons, 1996

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[4]

R. L. Wilson, Polynomial equations over matrices , Rutgers University, manuscript

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Engelking, General Topology, Heldermann Verlag, Berlin, 1989

R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989

1989

[2] [2]

Fuchs, A

D. Fuchs, A. Schwarz, A matrix Vieta Theorem, E. B. Dynkin Seminar, Amer. Math. Soc. Thansl. Ser. 2, 169 (1996)

1996

[3] [3]

Lutkepohl, Handbook of Matrices, John Wiley & Sons, 1996

H. Lutkepohl, Handbook of Matrices, John Wiley & Sons, 1996

1996

[4] [4]

R. L. Wilson, Polynomial equations over matrices , Rutgers University, manuscript