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Central polynomials of minimal degree for matrices

This paper investigates methods for finding low-degree central polynomials for matrix algebras and proves that $4\times 4$ matrices over a field of characteristic 0 possess neither central polynomials nor polynomial identities in two variables of degree 12 or less.

Original authors: Vesselin Drensky, Boyan Kostadinov

Published 2026-07-07

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Vesselin Drensky, Boyan Kostadinov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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 working with a giant, complex machine made of $4 \times 4$ grids of numbers (matrices). In this machine, the order in which you multiply things matters. If you swap the order, you usually get a different result.

However, there is a special "center" to this machine. Think of it as a control room where everything is perfectly balanced. If you put a specific object in this control room, it doesn't matter what other parts of the machine you touch; this object stays exactly the same. It commutes with everything.

Mathematicians are looking for a special "magic recipe" (a polynomial) that, when you feed it any set of matrices from this machine, always spits out an object that belongs to this control room. But here's the catch: the recipe must not be a "trick" that always outputs zero. It has to actually do something.

The Big Question:
How complicated does this magic recipe have to be? Specifically, how many "ingredients" (variables) and how many "steps" (degree) does it need?

For a long time, a mathematician named Formanek guessed the answer. He thought that for a $4 \times 4$ machine, the simplest possible recipe would need 13 steps. He had examples of recipes with 13 steps, but nobody knew if a simpler one (with 12 steps or fewer) existed.

What This Paper Does:
The authors, Drensky and Kostadinov, decided to play detective to see if a 12-step recipe could exist. They didn't just guess; they built a sophisticated search engine using two powerful tools:

Symmetry Patterns (Representation Theory): Instead of checking every single possible recipe (which would be like checking every grain of sand on a beach), they grouped recipes by their "symmetry shapes." This is like sorting a massive pile of Lego bricks by color and shape before trying to build something. It drastically reduces the number of things you need to check.
The "Upper Triangular" Filter: They used a clever trick involving a simpler type of matrix (one where all the numbers below the diagonal are zero). They proved that if a magic recipe exists for the big machine, it must also work in a specific way for this simpler machine. This allowed them to ignore huge chunks of impossible recipes right away.

The Investigation:
They focused on recipes that use only two variables (let's call them $x$ and $y$ ) and have a total length (degree) of 10, 11, or 12.

The Setup: They constructed a massive list of every possible "symmetry shape" a 12-step recipe could have.
The Test: They took these shapes and fed them into a test machine. They replaced the variables $x$ and $y$ with specific, generic $4 \times 4$ matrices.
The Result: For a recipe to be a "central polynomial," the output must be a scalar (a number on the diagonal) and nothing else. When they ran the math, they found that for every single possible shape they tested, the only way to make the output work was to set all the coefficients to zero.

The Conclusion:
In plain English: There is no magic recipe with 12 steps or fewer that works for $4 \times 4$ matrices.

If you try to build one, the math forces you to cancel everything out, leaving you with nothing. This proves that Formanek's guess was likely correct: the simplest possible recipe for a $4 \times 4$ matrix machine really does need 13 steps.

A Bonus Discovery:
While looking for these "magic recipes," they also checked if there were any "trick recipes" (polynomial identities) that always output zero for $4 \times 4$ matrices using only two variables and 12 steps or fewer. They found none of those either.

Why This Matters (According to the Paper):
This isn't just about counting steps. It confirms a deep pattern in how these mathematical machines work. The authors show that by combining symmetry patterns with specific algebraic filters, you can solve problems that would otherwise require checking billions of possibilities. They proved that for $4 \times 4$ matrices, the "minimal degree" of these central polynomials is indeed 13, closing the door on the possibility of a simpler 12-step solution.

Technical Summary: Central Polynomials of Minimal Degree for Matrices

Problem Statement
The paper addresses the determination of the minimal degree of central polynomials for the algebra of $n \times n$ matrices, $M_n(K)$ , over a field $K$ of characteristic 0. A polynomial $c(x_1, \dots, x_d) \in K\langle X \rangle$ is a central polynomial if its evaluation on $M_n(K)$ always yields a scalar matrix (an element of the center), yet it is not a polynomial identity (it does not vanish identically).

Formanek conjectured that this minimal degree is given by the formula:
$\text{mindeg}(M_n(K)) = \frac{1}{2}(n^2 + 3n - 2)$
This conjecture holds for $n \le 3$ (where the degrees are 1, 4, and 8, respectively). For $n=4$ , the conjectured minimal degree is 13. While central polynomials of degree 13 exist for $M_4(K)$ , it was an open question whether central polynomials of lower degree (specifically degree 12) exist. The paper focuses on resolving this case for polynomials in two variables.

Methodology
The authors employ a computational approach grounded in the representation theory of the general linear group $GL_d(K)$ and the symmetric group $S_m$ , combined with structural results from the theory of polynomial identity (PI) algebras.

Reduction to Multilinear and Proper Polynomials:
- The search is restricted to multilinear polynomials. If a central polynomial of degree $m$ does not exist, none exist for degrees $j < m$ .
- Using Lemma 3.6 (based on Maltsev's work on upper triangular matrices), the authors restrict the search to "proper" polynomials—linear combinations of products of commutators. Specifically, for $M_4(K)$ , any minimal degree central polynomial must be a linear combination of products of at least 4 commutators.
- The search is further narrowed to highest weight vectors of irreducible $GL_d(K)$ -modules $W(\lambda)$ , where $\lambda = (\lambda_1, \lambda_2)$ is a partition of the degree $m$ .
Representation Theoretic Filtering:
- The authors utilize the decomposition of tensor products of $GL_d(K)$ -modules to identify which partitions $\lambda$ can contain central polynomials.
- For degree 12 in two variables, they identify that only partitions $\lambda = (7, 5)$ and $\lambda = (6, 6)$ (with $\lambda_2 \ge 5$ ) are relevant candidates, based on the decomposition of products of commutators.
- This approach significantly reduces the number of unknowns compared to standard multilinear methods. For degree 12, instead of solving systems with hundreds of unknowns (as in pure $GL_d$ methods) or millions (as in full multilinear methods), the authors construct systems with only 60 and 31 unknowns, respectively.
Computational Verification:
- For each candidate highest weight vector $w_\lambda(x, y)$ , the authors substitute generic matrices. To simplify computation, they replace the first variable $x$ with a diagonal generic matrix $u = \text{diag}(u_1, u_2, u_3, u_4)$ and the second variable $y$ with specific constant matrices (e.g., $v, v_1, v_2$ ).
- They evaluate the resulting matrix $w_\lambda(u, v)$ . For the polynomial to be central, all off-diagonal entries must vanish. This generates a homogeneous linear system for the coefficients of the polynomial.
- The authors verify that these systems possess only the trivial solution, implying no non-zero central polynomials (or polynomial identities) exist for the specific partitions.

Key Contributions and Results

Resolution for $n=4$ : The paper proves that the algebra $M_4(K)$ does not possess central polynomials in two variables of degree $\le 12$ . Consequently, the minimal degree for central polynomials in two variables for $M_4(K)$ is at least 13.
Polynomial Identities: As a byproduct of the same computations, the authors prove that $M_4(K)$ does not have polynomial identities in two variables of degree $\le 12$ .
Efficiency of Method: The paper demonstrates that combining $GL_d(K)$ representation theory with PI-theory constraints (specifically the requirement for proper polynomials and the structure of identities in upper triangular matrices) is computationally superior to using representation theory alone or full multilinear expansion. For the degree 12 case, this reduced the problem to solving linear systems with 60 and 31 unknowns, compared to 297 and 132 unknowns using standard $GL_d$ methods.

Significance
The paper provides strong evidence supporting Formanek's conjecture for $n=4$ . While it does not prove the conjecture for all $n$ , it definitively settles the case for two-variable polynomials of degree 12. The work highlights the utility of computational methods in PI-algebra theory, showing that complex structural questions about matrix algebras can be resolved by reducing the search space through representation theory and then applying linear algebra on personal computers. The results confirm that the minimal degree for $M_4(K)$ is indeed 13 for the specific case of two variables, aligning with the quadratic formula $\frac{1}{2}(n^2 + 3n - 2)$ .

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