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arxiv: 2604.27592 · v1 · submitted 2026-04-30 · 🧮 math.RA · math.GR· math.NT

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Polynomial Maps with Constants on Matrix Algebra

Anupam Singh, Prachi Saini

Pith reviewed 2026-05-07 09:00 UTC · model grok-4.3

classification 🧮 math.RA math.GRmath.NT
keywords polynomial maps with constantsmatrix algebrassurjectivitynullitynoncommutative polynomialsimages of mapsM_n(F)
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The pith

For 3x3 and 4x4 matrices the map A1 X^k + A2 Y^k hits every matrix exactly when the nullity of A2 meets a condition depending on n and k, assuming A1 is invertible.

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

The paper studies maps from pairs of n by n matrices to a single n by n matrix given by evaluating the expression A1 times the k-th power of the first variable plus A2 times the k-th power of the second variable. Earlier results settled the n equals 2 case completely. Here the authors fix one coefficient matrix to be invertible and connect the surjectivity of the resulting map to the dimension of the kernel of the remaining coefficient. For the specific sizes n equals 3 and n equals 4 they obtain exact necessary and sufficient conditions phrased in terms of n, the exponent k, and that kernel dimension.

Core claim

Assuming A1 is invertible, the polynomial map ω(x1, x2) = A1 x1^k + A2 x2^k from M_n(F)^2 to M_n(F) is surjective if and only if the nullity of A2 satisfies a necessary and sufficient condition depending on n and k; this condition is determined completely when n equals 3 or 4.

What carries the argument

The evaluation map ω defined by the two-variable polynomial with constant coefficients A1 X^k + A2 Y^k, whose surjectivity on matrix pairs is reduced to linear-algebraic data on the kernel of A2.

If this is right

  • When the nullity condition holds the image of ω is the entire matrix algebra M_n(F).
  • When the nullity condition fails the image is a proper subset whose size or structure can be described from the same data.
  • The classification for n=3 and n=4 reduces the surjectivity question to a finite check on the possible nullities 0 through n.
  • The same reduction technique that links surjectivity to nullity(A2) applies whenever A1 is invertible, independent of the specific values of n=3 or 4.

Where Pith is reading between the lines

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

  • The same nullity criterion may continue to govern surjectivity for n greater than 4, though the paper stops at n=4.
  • Replacing the monomials X^k and Y^k by other words or adding more variables would test whether the nullity link persists beyond this two-term form.
  • The result supplies an explicit obstruction (high nullity of A2) that prevents the map from being onto, which can be checked directly on any concrete pair of matrices.

Load-bearing premise

One of the coefficient matrices, A1, is invertible.

What would settle it

For n=3 and a fixed k, pick a matrix A2 whose nullity lies on the boundary of the claimed condition and check whether every 3 by 3 matrix appears as A1 B^k + A2 C^k for some B and C; a single matrix that is missed (or hit contrary to the prediction) falsifies the classification.

read the original abstract

Let $\mathcal A$ be an $\mathbb F$-algebra and $\omega \in \mathcal A\langle x_1, \ldots, x_m \rangle$ which defines a map $\mathcal A^m \rightarrow \mathcal A$ by evaluation, called a polynomial map with constant. We consider $\mathcal {A} = M_n(\mathbb{F})$, the algebra of $n \times n$ matrices over an algebraically closed field $\mathbb{F}$ of characteristic $0$, and polynomial maps given by $\omega(x_1, x_2) = A_1x_1^k + A_2x_2^k$, where $A_1,A_2\in M_n(\mathbb F)$. For $n=2$, the images of such a map is competely determined in an earlier work (Panja, S.; Saini, P.; Singh, A., Images of polynomial maps with constants, Mathematika 71 (2025), no. 3, Paper No. e70031). In this article, by assuming one of the coefficients, say $A_1$, is invertible, we relate the surjectivity of $\omega$ to the nullity of $A_2$. When $n=3, 4$, we completely classify the surjectivity of $\omega(x_1, x_2)$ by obtaining the necessary and sufficient condition in terms of $n$, $k$, and the nullity of $A_2$.

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 studies polynomial maps with constants on the matrix algebra M_n(F), where F is algebraically closed of characteristic zero. It focuses on maps of the form ω(x1, x2) = A1 x1^k + A2 x2^k with A1, A2 in M_n(F). For n=2 the image is already classified in prior work by overlapping authors; the new contribution assumes A1 invertible (so that B = A1^{-1} A2 has the same nullity as A2) and claims a complete classification of surjectivity for n=3 and n=4 by necessary-and-sufficient conditions expressed solely in terms of n, k, and nullity(A2).

Significance. If the classification is correct and the reduction to nullity alone is justified, the result would give an explicit, computable criterion for surjectivity of these two-term polynomial maps on small matrix algebras, extending the n=2 case and potentially serving as a test case for broader questions about images of non-linear maps on M_n(F).

major comments (2)
  1. [Abstract; statements of the main theorems for n=3 and n=4] The central claim (abstract and the statements for n=3,4) asserts that surjectivity depends only on nullity(A2) once A1 is invertible. However, the set of k-th powers is not a vector space, and the linear span of {X^k + B Y^k} can depend on the Jordan form (or minimal polynomial) of B even when rank(B) is fixed. The manuscript must supply an explicit argument showing that all matrices of a given nullity produce the same image; without it the classification is incomplete.
  2. [Section 2 (preliminaries and reduction)] The reduction step that replaces A2 by B = A1^{-1} A2 (under the standing assumption that A1 is invertible) is used throughout the classification. It is not shown that this conjugation preserves the image of the map for every k; a short verification or counter-example check for small k would be needed to confirm the reduction is valid.
minor comments (2)
  1. [Abstract] Abstract: 'competely' should be 'completely'.
  2. [Introduction] The notation for the polynomial map ω and the field F is introduced without an explicit reminder that F is algebraically closed of characteristic zero; a single sentence in the introduction would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address each major comment below and will incorporate the necessary clarifications and arguments into the revised version to strengthen the presentation of the classification.

read point-by-point responses

  1. Referee: [Abstract; statements of the main theorems for n=3 and n=4] The central claim (abstract and the statements for n=3,4) asserts that surjectivity depends only on nullity(A2) once A1 is invertible. However, the set of k-th powers is not a vector space, and the linear span of {X^k + B Y^k} can depend on the Jordan form (or minimal polynomial) of B even when rank(B) is fixed. The manuscript must supply an explicit argument showing that all matrices of a given nullity produce the same image; without it the classification is incomplete.

    Authors: We agree that an explicit argument is required to confirm that surjectivity depends only on the nullity of A2 (equivalently, the rank of B) and is independent of the Jordan form. In preparing the classification for n=3 and n=4, we performed case-by-case analysis over the possible ranks, using the structure of matrix k-th powers in low dimensions. To address the concern directly, we will add a new lemma in Section 2 (or immediately preceding the main theorems) that explicitly verifies the independence: for each fixed nullity in dimensions 3 and 4, we enumerate the possible Jordan canonical forms of B, compute the image of the map x^k + B y^k in each case, and show that the resulting image coincides. This will make the reduction to nullity fully rigorous. revision: yes

  2. Referee: [Section 2 (preliminaries and reduction)] The reduction step that replaces A2 by B = A1^{-1} A2 (under the standing assumption that A1 is invertible) is used throughout the classification. It is not shown that this conjugation preserves the image of the map for every k; a short verification or counter-example check for small k would be needed to confirm the reduction is valid.

    Authors: We appreciate this observation. The reduction is valid for any k because, when A1 is invertible, we have ω(x1, x2) = A1 x1^k + A2 x2^k = A1 (x1^k + B x2^k) with B = A1^{-1} A2. Left multiplication by the invertible matrix A1 is a bijective linear automorphism of the vector space M_n(F). Consequently, the image of ω equals the full matrix algebra if and only if the image of the reduced map x1^k + B x2^k is the full algebra; the argument is independent of k. We will insert a short, self-contained paragraph in Section 2 providing this verification, together with the explicit equivalence. revision: yes

Circularity Check

0 steps flagged

Minor self-citation for n=2 case; n=3,4 classification independent

full rationale

The paper cites prior overlapping-author work only to recall the n=2 case and then states its own assumption (A1 invertible) and derives a necessary-and-sufficient condition for surjectivity when n=3 or 4 expressed solely in terms of n, k and nullity(A2). No equation or step in the abstract or described derivation reduces the new classification to the cited result by definition or construction; the central claim for n=3,4 therefore retains independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper uses the standard definition of matrix algebra over an algebraically closed field of characteristic zero and the prior n=2 classification; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption F is an algebraically closed field of characteristic zero
    Standard setup stated in the abstract for the matrix algebra M_n(F).

pith-pipeline@v0.9.0 · 5569 in / 1263 out tokens · 47084 ms · 2026-05-07T09:00:54.063810+00:00 · methodology

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Reference graph

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