Polynomial Maps with Constants on Matrix Algebra

Abstract: Let $\mathcal A$ be an $\mathbb F$-algebra and $ω\in \mathcal A\langle x_1, \ldots, x_m \rangle$ 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 $ω(x_1, x_2) = A_1x_1^k + A_2x_2<sup>k$, 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 $ω$ to the nullity of $A_2$. When $n=3, 4$, we completely classify the surjectivity of $ω(x_1, x_2)$ by obtaining the necessary and sufficient condition in terms of $n$, $k$, and the nullity of $A_2$.

• The paper establishes necessary and sufficient surjectivity criteria for polynomial maps with non-scalar matrix constants based on the nilpotent structure of A2.
• It employs algebraic reductions and Jordan canonical forms to derive sharp bounds on surjectivity, particularly demonstrating complete classification for n=3 and n=4.
• The analysis clarifies how the count of nilpotent Jordan blocks directly governs the image of the map, bridging results in matrix algebra and noncommutative polynomial mappings.

Polynomial Maps with Constants on Matrix Algebra: Surjectivity Criteria and Low-Dimensional Classification

Introduction

The paper "Polynomial Maps with Constants on Matrix Algebra" (2604.27592) investigates the surjectivity of polynomial maps with non-scalar constants on matrix algebras, specifically $M_n(\mathbb{F})$ where $\mathbb{F}$ is an algebraically closed field of characteristic zero. The principal object of study is maps of the form $\omega(x_1, x_2) = A_1 x_1^k + A_2 x_2^k$, where $A_1, A_2 \in M_n(\mathbb{F})$ and $A_1$ is invertible. The paper provides a rigorous classification of surjectivity for these maps, especially for $n = 3, 4$, in terms of the nilpotency structure of $A_2$, specifically the number of its nilpotent Jordan blocks.

General Surjectivity Criterion

The central result is a necessary and sufficient criterion for the surjectivity of the polynomial map $\omega(x_1, x_2)$ based on the nullity of $A_2$, denoted $r_0$, which is the number of Jordan blocks of $\mathbb{F}$0 corresponding to the eigenvalue $\mathbb{F}$1. The main theorem asserts:

• Surjectivity holds if $\mathbb{F}$2: Regardless of $\mathbb{F}$3 or $\mathbb{F}$4, if the nullity of $\mathbb{F}$5 is at most one, for any invertible $\mathbb{F}$6, the map $\mathbb{F}$7 is surjective on $\mathbb{F}$8.
• Surjectivity fails if $\mathbb{F}$9: When $\omega(x_1, x_2) = A_1 x_1^k + A_2 x_2^k$0, surjectivity is lost if the dimension $\omega(x_1, x_2) = A_1 x_1^k + A_2 x_2^k$1 is insufficient relative to $\omega(x_1, x_2) = A_1 x_1^k + A_2 x_2^k$2 and $\omega(x_1, x_2) = A_1 x_1^k + A_2 x_2^k$3.

This characterization is explicitly derived from an algebraic reduction involving conjugation invariance and Jordan canonical forms, leveraging the structure of matrix algebra over algebraically closed fields. The key insight is associating surjectivity to the ability to construct enough independent solutions given the nilpotent structure imposed by $\omega(x_1, x_2) = A_1 x_1^k + A_2 x_2^k$4.

Low-Dimensional Complete Classification

For $\omega(x_1, x_2) = A_1 x_1^k + A_2 x_2^k$5 and $\omega(x_1, x_2) = A_1 x_1^k + A_2 x_2^k$6, the paper provides a full set of necessary and sufficient conditions:

• For $\omega(x_1, x_2) = A_1 x_1^k + A_2 x_2^k$7: $\omega(x_1, x_2) = A_1 x_1^k + A_2 x_2^k$8 is surjective if and only if $\omega(x_1, x_2) = A_1 x_1^k + A_2 x_2^k$9.
• For $A_1, A_2 \in M_n(\mathbb{F})$0: $A_1, A_2 \in M_n(\mathbb{F})$1 is surjective if and only if $A_1, A_2 \in M_n(\mathbb{F})$2.

These results are established via explicit analysis of Jordan block configurations and constructive proofs. When surjectivity fails (e.g., $A_1, A_2 \in M_n(\mathbb{F})$3, $A_1, A_2 \in M_n(\mathbb{F})$4 for $A_1, A_2 \in M_n(\mathbb{F})$5), the image is characterized precisely: the map misses matrices possessing a certain nilpotent block structure. The arguments rely on careful rank and eigenvalue considerations, and exploit known results about powers of nilpotent Jordan blocks (citing Miller) to frame non-surjectivity via block size constraints.

Sharp and Strong Claims

A particularly strong assertion is the sharpness of the surjectivity bounds: if $A_1, A_2 \in M_n(\mathbb{F})$6 and $A_1, A_2 \in M_n(\mathbb{F})$7, surjectivity cannot hold; thus, the criterion reflects a boundary phenomenon. The work thoroughly classifies the image sets in exceptional cases, identifying the precise algebraic form of matrices not included in the image.

Additionally, the approach and reduction technique generalize prior results on scalar-coefficient polynomial maps, extending the framework to non-scalar constants and higher-degree Waring-type polynomials. The results affirm that invertibility of at least one coefficient is often sufficient for surjectivity, but nilpotent structure introduces sharp obstructions.

Theoretical and Practical Implications

From a theoretical standpoint, the classification leverages the interplay between polynomial evaluation, Jordan forms, and matrix algebraic structure, framing surjectivity in terms of explicit algebraic invariants. The results situate polynomial maps with constants as natural analogues of word maps with constants in group theory, connecting algebraic surjectivity phenomena across associative and group-theoretic contexts.

Practically, understanding the surjectivity of such maps has implications for matrix representation theory, noncommutative polynomial mapping, and algebraic approaches to phenomena such as invertibility, spectra, and canonical decomposition. The sharp criteria can guide algorithmic approaches and symbolic manipulation in computer algebra systems for solving matrix equations of polynomial form.

Future Directions

This work points toward several future lines of inquiry:

• Extension to higher $A_1, A_2 \in M_n(\mathbb{F})$8 and multi-variable polynomial maps, exploring whether the sharp boundary observed for $A_1, A_2 \in M_n(\mathbb{F})$9 persists in larger matrix algebras.
• Investigation of surjectivity criteria under alternate field assumptions (e.g., real, finite fields) or in positive characteristic.
• Expansion to broader classes of polynomial maps, including multilinear and mixed-degree forms, and their behavior on central simple algebras.
• Analytic study of the distribution of images, completeness, and density of such mappings in matrix algebra.

The structural connection to group word maps, block theory, and Waring-type problems suggests that the frameworks developed here can be adapted to parallel problems in other algebraic systems.

Conclusion

The paper rigorously establishes surjectivity criteria for polynomial maps with constants on matrix algebras, providing sharp bounds and full classification for low-dimensional cases. The main results link surjectivity to nilpotent structure, codifying precise algebraic obstructions and demonstrating the power of canonical form reductions and block analysis. The implications reach across algebraic theory and computational practice, and the results lay a foundation for continued exploration of structural polynomial mapping in associative algebras.