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Simple polynomial equations over (mxm)-matrices

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Pith's one-line read For real 3x3 matrices, the solution set of any scalar-coefficient polynomial equation is a finite union of conjugation orbits of Jordan forms, and its covering dimension is always 0, 4, or 6.

desk verdict Genuinely new and sound m=3 classification of solution sets for scalar polynomial matrix equations; the only gaps are minor and concern the m>=4 appendages. read the letter →

arxiv 2507.07085 v1 pith:O5UV6ZC3 submitted 2025-07-09 math.RA math.GN

Vitalij A. Chatyrko , Alexandre Karassev This is my paper
classification math.RAmath.GN MSC 15A2454F4515B30
keywords polynomialmatrixequationscoveringdimensionrealJordanformconjugationorbitsolutionsetcentralizersigma-compact
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 settles, for real 3x3 matrices, what the solution set of a polynomial equation with scalar coefficients looks like. Any equation of the form X^n + a_{n-1}$X^{{n-1}}$ + ... + a_0 I = O has a solution set that is either empty or a disjoint union of conjugation orbits of real Jordan forms, each orbit determined by which roots of the scalar polynomial f_s(x)=x^n+...+a_0 are real, repeated, or complex. The covering dimension of each orbit is computed from the dimension of the centralizer of the corresponding Jordan form, and the dimension of the whole solution set is the maximum of the orbit dimensions: 0, 4, or 6 (or -1 if no solution exists). The same orbit decomposition is shown to hold for any m≥4, with the solution set still σ-compact and its dimension the maximum of finitely many orbit dimensions. A sympathetic reader would care because it reduces a nonlinear matrix equation to root-pattern bookkeeping and gives exact topological dimensions rather than only finiteness or boundedness statements.

What carries the argument

The machinery is the conjugation action of GL_m(R) on M_m(R): each solution set is a union of orbits Conj_B(GL_m(R)) = {$X^{{-1}}$BX}. The dimension of such an orbit is computed by Corollary 2.3 as dim GL_m(R) - dim G_B, where G_B is the centralizer (stabilizer) of B, a closed subgroup; this reduces a topological dimension question to linear algebra. For each possible real 3x3 Jordan form the centralizer is written out explicitly in Proposition 3.1, giving orbit dimensions 6, 4, and 0. The second ingredient is Lemma 4.3, which evaluates F_s on each Jordan block and shows that the only obstruction to being a solution is the vanishing of f_s and its derivatives at the eigenvalues.

What would settle it

Take the polynomial equation $X^{2}$ = O over 3x3 real matrices and compute the covering dimension of its solution set. The paper predicts 4 (Example 5.3: the union of the scalar-zero orbit and the orbit of the Jordan block J_1(0,0,0)). A direct parameter count showing the solution space has dimension other than 4, or exhibiting a solution not similar to one of the two listed Jordan forms, would refute the classification.

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Extended reading notes

Core claim

The central discovery is that for m=3 the solution set S of (2) is a disjoint union of sets S(J)=Conj_J(GL_3(R)) where J ranges over the real Jordan canonical forms whose diagonal blocks match the root pattern of f_s. Theorem 4.5 lists these: distinct real roots give S(J(a,b,c)) (dimension 6); a repeated real root with a different root gives S(J(a,a,b)) (dimension 4) or S(J_1(a,a,b)) (dimension 6, when the repeated root has multiplicity at least 2); a scalar J(a,a,a) gives a singleton (dimension 0); J_1(a,a,a) (dimension 4) and J_2(a,a,a) (dimension 6) appear when the root has multiplicity at least 2 or 3; and complex conjugate roots p±iq paired with a real root a give S(J_c(a,p,q)) (dimension 6), which is exactly the real form of the diagonal complex matrix U(a,p,q). Corollary 4.7 then states that dim S is -1 (empty) or the maximum of the orbit dimensions, hence 0, 4, or 6. For m≥4 the same logic yields a finite disjoint union of conjugation orbits of real Jordan forms, and Proposition 6.1 gives dim S = $m^{2}$ - m when all roots are distinct and the polynomial degree is at least m.

Load-bearing premise

The argument relies on the fact that the dimension of the quotient GL_3(R)/G_J equals dim GL_3(R) minus dim G_J for the centralizer subgroups G_J; if that additivity failed for any of the listed centralizers, the orbit dimensions 6, 4 and 0 would be wrong.

Editorial extensions

If this is right

  • For any scalar polynomial f_s of degree n≥2 and m=3, the solution set of F_s(X)=O is either empty or a finite disjoint union of at most seven types of conjugation orbits, each type tied to a root pattern of f_s.
  • The covering dimension of every such solution set is one of -1, 0, 4, 6, so the examples in Section 5 exhaust the possible dimensions.
  • The same orbit decomposition is valid for all m≥4, and when f_s has only distinct real roots and degree at least m the dimension is exactly m^2 - m.
  • When f_s has only non-real roots and m is odd, the solution set is empty (dimension -1), since a real odd-dimensional matrix must have a real eigenvalue.
  • The solution set S is σ-compact for every m, being a finite union of continuous images of GL_m(R).

Reading between the lines

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

  • The same centralizer-dimension method should extend to quaternionic or complex matrix equations, where orbit dimensions would depend on the base field and on the analogue of real Jordan forms.
  • For m≥4, the possible values of dim S are not classified; the paper's Question 7.1 invites a conjecture that the attainable dimensions form an arithmetic progression related to partitions of m.
  • One could test the classification computationally for small m by sampling random matrix solutions and checking conjugacy to the listed Jordan forms, which would expose any missing orbit types.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper studies the solution set S of the scalar-coefficient polynomial matrix equation X^n + a_{n-1}X^{n-1} + ... + a_0 I = O over real m x m matrices. For m = 3, it proves that S is either empty or a finite union of conjugacy orbits of real Jordan forms of seven types, with the admissibility conditions expressed in terms of the roots of the scalar polynomial f_s(x) and their multiplicities (Theorem 4.5). It then computes the dimension of each such orbit as 6, 4, or 0 by subtracting the dimension of the corresponding centralizer from dim GL_3(R) (Corollary 4.6), and concludes that dim S is the maximum of these dimensions (Corollary 4.7). Several explicit examples illustrate the possible values 0, 4, and 6. Section 6 sketches the structure for m >= 4 and states two propositions without proof.

Significance. The m = 3 classification and dimension formula are correct and self-contained. The Jordan-form argument is standard, the displayed centralizer computations in Proposition 3.1 check out, and the orbit dimensions agree with direct tangent-space counts. The paper is deliberately explicit: Lemma 4.3 records the polynomial evaluation on each Jordan form, Corollary 4.6 gives the orbit dimensions, and the examples in Section 5 make the result easy to verify. The main result is a natural extension of the authors' earlier m = 2 paper and answers Question 1.2 for m = 3. The m >= 4 part is only sketched, which limits the scope but does not affect the validity of the m = 3 contribution.

minor comments (5)
  1. [Corollary 2.3] The equality dim Conj_B(GL_m(R)) = dim GL_m(R)/G_B uses the fact that the continuous bijection alpha_B is a homeomorphism, or at least dimension-preserving. This is standard for the conjugation action of a Lie group, but it should be stated explicitly, since a continuous bijection alone need not preserve covering dimension.
  2. [Theorem 4.5] The theorem describes S as a disjoint union, but if (a,b,c) ranges over all ordered triples of distinct real roots, the sets S(J(a,b,c)) for permutations coincide, and S(J(a,a,b)) coincides with S(J(b,b,a)) for a != b; hence the same orbit is listed multiple times. Please either specify one representative per similarity class or replace 'disjoint union' by 'union'. The dimension conclusion in Corollary 4.7 is unaffected.
  3. [Section 6, Propositions 6.1 and 6.2] These propositions are stated without proof. If they are intended as results of the paper, proofs should be supplied; otherwise they should be rephrased as remarks or conjectures. This does not affect the m = 3 results.
  4. [Introduction and Example 5.2] The introduction defines the equation with degree n >= 2, but the abstract allows n >= 1 and Example 5.2(ii) uses X - I = O, which has degree 1. Please harmonize the standing assumption on n.
  5. [References] The reference [W] is listed only as a manuscript with no year; please complete the bibliographic information if it is publicly available.

Circularity Check

0 steps flagged · score 0.0 of 10

Self-contained Jordan-form classification; no circularity found.

full rationale

The m=3 result is derived from first principles rather than from fitted inputs or self-citation. Theorem 4.5 follows from the Jordan normal form classification (Proposition 4.1), the explicit evaluation formulas for Fs on Jordan blocks (Lemma 4.3), and similarity invariance (Lemma 4.4). The orbit dimensions in Corollary 4.6 are computed as dim GL3(R) − dim G_J via Proposition 3.1 and Corollary 2.3, where the quotient-dimension equality is cited from the external source [P], not from the authors' prior work. Corollary 4.7 then applies the countable sum theorem to a finite union of σ-compact conjugation orbits, with monotonicity giving the lower bound, so the maximum formula is not circular. The only references to the authors' earlier paper [ChK] are motivational (Question 1.2) and an analogue for an upper bound (Proposition 1.1); they are not load-bearing for the m=3 classification or dimension computation. No 'prediction' is renamed from a fitted parameter, and no uniqueness or structural premise is imported from a self-citation. The unproved Propositions 6.1 and 6.2 concern m ≥ 4 and do not affect the central m=3 claim. Hence the derivation is self-contained and no significant circularity is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper relies entirely on standard theorems from linear algebra and dimension theory. No free parameters are fitted to data, and no new entities are postulated. The central claim rests on the correctness of the Jordan form classification and the quotient dimension formula, both well-established.

assumptions (4)
  • standard math Every real 3x3 matrix is similar over R to one of the seven real Jordan forms listed in Proposition 4.1.
    This is the real Jordan canonical form theorem for 3x3 matrices, invoked in Proposition 4.1.
  • standard math For a locally compact group G and a closed subgroup H, dim G = dim H + dim(G/H).
    Cited as [P, Theorem 2] and used in Corollary 2.3 to compute the dimension of conjugation orbits as dim GL_3(R) - dim G_B.
  • standard math The countable sum theorem for covering dimension: a countable union of closed sets of dimension at most n has dimension at most n.
    Cited as [E, Theorem 7.2.1] and used in Corollary 4.7 to conclude that dim S is the maximum of the dimensions of the orbits, since each orbit is sigma-compact.
  • standard math A matrix X satisfies f_s(X)=O if and only if the minimal polynomial of X divides f_s.
    Implicitly used in Lemmas 4.3-4.4 to translate root conditions on the scalar polynomial into conditions on which Jordan blocks may appear in a solution.

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Pith. "Pith review of Simple polynomial equations over (mxm)-matrices." pith.science (2026). https://pith.science/paper/O5UV6ZC3

@misc{pith2026250707085,
  author       = {Pith},
  title        = {Pith review of: Simple polynomial equations over (mxm)-matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O5UV6ZC3}},
  note         = {Machine review of arXiv:2507.07085}
}
abstract

Let $m$ be any integer $\geq 3$. 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 $(m \times m)$-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 $n \geq 1$. We discuss its solution set $S$ supplied with the natural Euclidean topology. In particular, we describe the solution set $S$ for $m=3$ and calculate its dimension.

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Works this paper leans on

8 extracted references · 7 canonical work pages

  1. [1]

    V. A. Chatyrko, A. Karassev, Simple polynomial equations over (2 2) -matrices, arXiv:2506.07689

    work page Pith review arXiv

  2. [2]

    Engelking, General Topology, Heldermann Verlag, Berlin, 1989

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

    1989

  3. [3]

    Fuchs, A

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

    work page 1996

  4. [4]

    Gelfand, On the number of solutions of a quadratic equation, In: Globus: General Mathematical Seminar, 1, Independent University of Moscow, Moscow, 2004, 124-133 (in Russian)

    S. Gelfand, On the number of solutions of a quadratic equation, In: Globus: General Mathematical Seminar, 1, Independent University of Moscow, Moscow, 2004, 124-133 (in Russian)

    work page 2004

  5. [5]

    B. A. Pasynkov, On the coincidence of various definitions of dimensionality for factor spaces of locally bicompact groups, Uspekhi Matematicheskikh Nauk, 1962, 17(5), 129–135

    work page 1962

  6. [6]

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

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

    work page 1996

  7. [7]

    Slusky, Zeros of (2 2) -matrix polynomials, Comm

    M. Slusky, Zeros of (2 2) -matrix polynomials, Comm. Algebra, 2010, 38(11), 4212-4223

    work page 2010

  8. [8]

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

    work page

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