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Polynomial Maps with Constants on Matrix Algebra

This paper extends previous results on polynomial maps with constants on $2 \times 2$ matrices to $3 \times 3$ and $4 \times 4$ matrices by establishing necessary and sufficient conditions for the surjectivity of maps of the form $\omega(x_1, x_2) = A_1x_1^k + A_2x_2^k$ (where $A_1$ is invertible) in terms of the matrix size $n$ , the exponent $k$ , and the nullity of $A_2$ .

Original authors: Prachi Saini, Anupam Singh

Published 2026-05-01

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Prachi Saini, Anupam Singh

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are in a giant, infinite kitchen where the ingredients are matrices (grids of numbers). In this kitchen, you have a special recipe called a polynomial map.

Usually, recipes are simple: "Take some numbers, square them, add them up." But in this paper, the authors are cooking with a twist. They are using polynomial maps with constants. Think of this as a recipe where you don't just add ingredients; you also have to mix in some pre-existing, fixed "secret sauces" (the constants $A_1$ and $A_2$ ) before you even start cooking.

The specific recipe they are testing looks like this:
$\text{Result} = A_1 \times (\text{Ingredient}_1)^k + A_2 \times (\text{Ingredient}_2)^k$

Here, $A_1$ and $A_2$ are fixed matrices (the secret sauces), and you can choose any two matrices as your ingredients ( $x_1$ and $x_2$ ). The question is: Can you cook up every possible matrix in the kitchen using this recipe?

If you can make every single matrix, the recipe is surjective (it covers the whole menu). If there are some dishes you simply cannot make, no matter how hard you try, the recipe is not surjective.

The Main Characters

The Field ( $F$ ): Imagine this as the type of water you use. The authors assume it's "algebraically closed" (like a magical water that can dissolve any number problem) and has "characteristic 0" (standard math rules, no weird modular arithmetic).
The Secret Sauce $A_1$ : The authors assume this sauce is invertible. In our kitchen analogy, this means $A_1$ is a "super-sauce" that never ruins the dish; it's powerful enough to be reversed or undone. It's the reliable base of the recipe.
The Secret Sauce $A_2$ : This is the tricky one. It might be a "weak" sauce. The authors look at its nullity ( $r_0$ $r_{0}$ ).
- The Metaphor: Imagine $A_2$ $A_{2}$ as a sponge.
  - If the sponge is full of holes (high nullity), it absorbs a lot of the "power" of the recipe. It creates "dead zones" where you can't reach certain dishes.
  - If the sponge is solid (low nullity), it doesn't block much, and you can still reach almost everything.
- Mathematically, the "holes" in the sponge correspond to Jordan blocks of size 1 associated with the number 0. The more of these blocks, the "spongier" the matrix is.

The Big Discovery: The "Size vs. Sponge" Rule

The authors found a simple rule that predicts whether you can cook every dish or if you'll get stuck. It depends on three things:

$n$ : The size of your kitchen (the matrix dimension, e.g., $3 \times 3$ or $4 \times 4$ ).
$k$ : The power in the recipe (e.g., squaring, cubing).
$r_0$ : The "sponginess" (nullity) of the second sauce $A_2$ .

The Rule:
You can cook every dish (the map is surjective) if and only if:
$n > k \times (r_0 - 1)$

Let's break this down with an analogy:

$n$ is the size of your canvas.
$k \times (r_0 - 1)$ is the size of the "blind spot" created by the spongy sauce $A_2$ .
If your canvas ( $n$ ) is larger than the blind spot, you can paint the whole picture.
If your canvas is smaller than or equal to the blind spot, there are parts of the picture you can never reach.

What They Found for Small Kitchens ( $n=3$ and $n=4$ )

The authors tested this rule specifically for $3 \times 3$ and $4 \times 4$ matrices and found it works perfectly.

If $A_2$ is solid ( $r_0 = 0$ or $1$): The blind spot is tiny or non-existent. You can make every matrix, no matter how big the kitchen is.
If $A_2$ is very spongy ( $r_0$ is high): The blind spot grows.
- Example: In a $3 \times 3$ kitchen ( $n=3$ ) with a cubing recipe ( $k=3$ ), if your sponge has 2 holes ( $r_0=2$ ), the blind spot size is $3 \times (2-1) = 3$ . Since the kitchen size (3) is not greater than the blind spot (3), you cannot make every dish. There will be specific matrices you simply cannot create.

The "Missing Dishes"

When the recipe fails (is not surjective), the authors didn't just say "it fails." They described exactly what is missing.

For the $3 \times 3$ case, the missing dishes are a very specific type of matrix: ones that look like a small, nilpotent block (a matrix that becomes zero if you square it) sitting in the corner, surrounded by zeros.
It's like saying, "You can cook every steak, every salad, and every soup, except for the specific dish that is a perfectly square, zero-tasting block of tofu."

Summary

This paper is about understanding the limits of a specific mathematical recipe.

The Setup: You have a recipe mixing two variable ingredients with two fixed sauces.
The Condition: One sauce is strong; the other might be "spongy" (have a high nullity).
The Result: Whether you can make everything depends on a simple inequality: Is the kitchen bigger than the sponge's blind spot?
- Yes? You can make everything.
- No? There are specific, predictable dishes you can never make.

The authors proved that for small kitchens ( $3 \times 3$ and $4 \times 4$ ), this rule is the absolute truth, giving a complete map of what is possible and what is impossible.

1. Problem Statement

The paper investigates the surjectivity of a specific class of polynomial maps defined on the matrix algebra $M_n(F)$ , where $F$ is an algebraically closed field of characteristic 0.

The map under consideration is:
$\omega(x_1, x_2) = A_1 x_1^k + A_2 x_2^k$
where:

$x_1, x_2 \in M_n(F)$ are the variables.
$k \geq 2$ is an integer.
$A_1, A_2 \in M_n(F)$ are fixed non-zero coefficient matrices.
Crucial Assumption: $A_1$ is invertible ( $A_1 \in GL_n(F)$ ).

The central question is: For which pairs $(A_1, A_2)$ is the map $\omega$ surjective onto $M_n(F)$ ?

This problem extends the classical L'vov-Kaplansky conjecture (which concerns multilinear polynomials without constants) and Waring-type problems (representing elements as sums of powers) to the setting of "polynomial maps with constants." Previous work had addressed the case $n=2$ and specific cases for $n=3,4$ with scalar coefficients. This paper generalizes the analysis to non-scalar coefficients where at least one is invertible.

2. Methodology

The authors employ a combination of linear algebra, Jordan canonical form theory, and constructive matrix analysis. The methodology proceeds through several reduction steps:

Conjugation Invariance: The property of surjectivity is invariant under simultaneous conjugation. If $g \in GL_n(F)$ , then $\omega$ is surjective if and only if $\omega_g(x_1, x_2) = (gA_1g^{-1})x_1^k + (gA_2g^{-1})x_2^k$ is surjective.
Reduction to a Single Matrix: Since $A_1$ is invertible, the map $\omega$ is surjective if and only if the map $\tilde{\omega}(x_1, x_2) = x_1^k + B x_2^k$ is surjective, where $B = A_1^{-1}A_2$ .
Jordan Canonical Form: Since $F$ is algebraically closed, $B$ can be assumed to be in Jordan canonical form. The problem reduces to analyzing the structure of $B$ , specifically the number of Jordan blocks corresponding to the eigenvalue 0.
Nullity and Nilpotent Blocks: Let $r_0$ denote the nullity of $A_2$ (which equals the nullity of $B$ ). By Lemma 2.1, $r_0$ is exactly the number of Jordan blocks of $B$ associated with the eigenvalue 0.
Case Analysis:
- Case $r_0 \leq 1$ : The authors prove surjectivity by constructing solutions. If $r_0=0$ , $B$ is invertible. If $r_0=1$ , $B$ has exactly one nilpotent block. They use specific constructions involving row spaces and invertibility of submatrices to show that any target matrix $C$ can be written as $X_1^k + B X_2^k$ .
- Case $r_0 \geq 2$ : The authors establish a necessary condition for surjectivity. They utilize a result by Miller regarding the size of Jordan blocks in the $k$ -th power of a nilpotent matrix. They construct a specific counter-example matrix $C$ that cannot be in the image if the dimension $n$ is too small relative to $k$ and $r_0$ .

3. Key Contributions and Results

A. General Necessary and Sufficient Conditions (Theorem 1.2)

The paper establishes a fundamental relationship between the dimension $n$ , the power $k$ , and the nullity $r_0$ of $A_2$ :

Sufficiency: If $r_0 \leq 1$ , the map is always surjective, regardless of $n$ and $k$ .
Necessity: If $n \leq k(r_0 - 1)$ , the map is not surjective.

B. Complete Classification for Low Dimensions (Theorem 1.3)

For $n=3$ and $n=4$ , the authors prove that the condition derived in the necessity part is also sufficient. Specifically, for $n \in \{3, 4\}$ :
$\omega \text{ is surjective} \iff n > k(r_0 - 1)$

This leads to explicit classifications presented in Tables 1 and 2:

For $n=3$ :
- Surjective if $r_0 = 0$ or $r_0 = 1$ .
- Surjective if $r_0 = 2$ and $k=2$ (since $3 > 2(2-1)$ ).
- Not surjective if $r_0 = 2$ and $k \geq 3$ (since $3 \leq k(1)$ ).
- Image Description: When not surjective, the image misses a specific set of matrices involving a $2 \times 2$ nilpotent block.
For $n=4$ :
- Surjective if $r_0 \leq 1$ .
- Surjective if $r_0 = 2$ and $k \in \{2, 3\}$ (since $4 > k(1)$ ).
- Not surjective if $r_0 = 2$ and $k \geq 4$ , or if $r_0 = 3$ (regardless of $k \geq 2$ ).

C. Structural Insight

The paper demonstrates that the nilpotent structure of the coefficient matrix $A_2$ (specifically the number of zero-eigenvalue Jordan blocks) is the decisive factor for surjectivity, rather than just the rank or determinant. The "obstruction" to surjectivity arises when the target matrix requires a Jordan block structure that cannot be generated by the $k$ -th power of a matrix given the constraints imposed by the nilpotent part of $A_2$ .

4. Significance

Generalization of Word Maps: The work bridges the gap between group theory (word maps) and associative algebra theory, extending Waring-type problems to maps with non-scalar constants.
Resolution of Low-Dimensional Cases: It provides a complete solution for $n=3$ and $n=4$ , filling a gap left by previous work which had only solved $n=2$ .
Sharpness of Bounds: The results suggest that the inequality $n > k(r_0 - 1)$ is likely the optimal condition for surjectivity for all $n$ , leading the authors to propose this as a conjecture for the general case (Question 6.1).
Constructive Proofs: Unlike non-constructive existence proofs, the paper provides explicit constructions for $X_1$ and $X_2$ in the surjective cases, detailing how to manipulate row spaces and Jordan forms to achieve the target matrix.

5. Conclusion

The paper successfully characterizes the surjectivity of polynomial maps with constants on matrix algebras when one coefficient is invertible. It identifies the nullity of the second coefficient as the critical parameter. The main finding is that surjectivity holds if and only if the matrix dimension $n$ is strictly greater than $k$ times the number of zero-eigenvalue Jordan blocks of the second coefficient minus one ( $n > k(r_0 - 1)$ ). This result unifies and extends previous findings in the field of noncommutative polynomial maps.