$m$-nil-clean nonderogatory matrices

Technical Summary: m-Nil-Clean Nonderogatory Matrices

Problem Statement
The paper investigates the decomposition of nonderogatory (cyclic) matrices over a field $F$ of positive characteristic $p$. Specifically, it addresses the problem of expressing an $n \times n$ nonderogatory matrix $A$ as a sum of $m$ idempotent matrices ($E_1, \dots, E_m$) and a nilpotent matrix $N$, such that $A = \sum_{i=1}^m E_i + N$. This decomposition is termed an "$m$-nil-clean" decomposition. The study focuses on determining the existence of such decompositions for matrices with traces in the set $\{k \cdot 1_F \mid k \in \{0, 1, \dots, p-1\}\}$, and crucially, establishing tight upper bounds on the nilpotence index of the matrix $N$ (i.e., finding the smallest $k$ such that $N^k = 0$).

Methodology
The authors employ a structural approach centered on companion matrices, leveraging the fact that every nonderogatory matrix is similar to a companion matrix. The methodology proceeds through the following steps:

1. Reduction to Companion Matrices: The problem is reduced to analyzing companion matrices $C$. The authors utilize the "fitting lemma" (Lemma 2.2), which states that any companion matrix $C$ with trace $t$ is similar to $C' + D'$, where $C'$ is a trace-zero companion matrix and $D'$ is a diagonal matrix with trace $t$.
2. Decomposition up to a Diagonal: The authors introduce the concept of an "$m$-nil-clean decomposition up to a diagonal matrix $D_0$." This involves finding idempotents and a nilpotent such that $C = \sum E_i + N - D_0$.
3. Partitioning and Nilpotent Construction: To control the nilpotence index of $N$, the authors define "good partitions" and "trivial partitions" of the integer $n-1$.
   • A good partition of $n-1 = d_1 + \dots + d_r$ requires $d_i > 1$ for $i \ge 2$.
   • A trivial partition is $n-1 = d_1$.
   • Nilpotent matrices $N$ are constructed based on these partitions using block structures involving lower triangular Jordan cells ($J_k$) and specific zero-row constraints. This construction ensures that the nilpotence index of $N$ is bounded by $\max(d_1+1, d_2, \dots, d_r)$ for good partitions, or $n-1$ for trivial partitions.
4. Trace Matching: The authors construct specific idempotent matrices (referred to as "severance matrices" and diagonal idempotents) to ensure the sum of their traces can cover the necessary range of values to match the trace of the target matrix $C$. By adjusting the diagonal entries of these idempotents, they demonstrate that the set of possible traces for $\sum E_i$ forms a contiguous interval of integers modulo $p$.
5. Synthesis: By combining the trace-matching capability with the structural decomposition up to a diagonal, and applying the fitting lemma, the authors prove the existence of the exact $m$-nil-clean decomposition without the residual diagonal term.

Key Contributions and Results
The paper establishes the following main theorem (Theorem 3.3):

Let $F$ be a field of positive characteristic $p$, and let $m, n$ be positive integers with $m \ge 2$ and $n \le p \le mn - 1$. For every $n \times n$ nonderogatory matrix $A \in M_n(F)$ with trace in $\{k \cdot 1_F \mid k \in \{0, 1, \dots, p-1\}\}$, there exist $m$ idempotent matrices $E_1, \dots, E_m$ and a nilpotent matrix $N$ such that:
$$A = E_1 + E_2 + \dots + E_m + N$$
where $N^k = 0$. The bound $k$ on the nilpotence index depends on the relationship between $n, m, p$ and the parity of $n$:

• Case 1: If $p \in \{nm-1, nm-2\}$, then $k = n$.
• Case 2: If $p = nm-3$, then $k = n-1$.
• Case 3: Otherwise, let $r = \lfloor \frac{nm-p}{2} \rfloor$.
  • If $n$ is even, $k = \max(2, 1 + \lfloor \frac{n-1}{r} \rfloor)$.
  • If $n$ is odd, $k = \max(3, 1 + \lfloor \frac{n-1}{r} \rfloor)$.

Additionally, the paper provides a specific result for the case where $n > p$ (Remark 3.4). In this scenario, setting $m=2$, any such nonderogatory matrix $A$ is the sum of two idempotent matrices and:

• A square-zero matrix ($N^2=0$) if $n$ is even.
• A matrix with nilpotence index at most 3 ($N^3=0$) if $n$ is odd.

Significance and Claims
The paper positions itself as an extension of existing literature on nil-clean and clean decompositions. It cites previous work by Nicholson, Diesl, and others regarding nil-clean matrices over division rings and finite fields. The authors note that while previous results established the existence of nil-clean decompositions (sum of one idempotent and one nilpotent) or bounds on nilpotence indices (e.g., index $\le 4$ over $F_2$), this work generalizes the concept to $m$-nil-clean decompositions for nonderogatory matrices over fields of arbitrary positive characteristic.

The significance claimed is the precise determination of the nilpotence index bound for these decompositions. The authors demonstrate that by carefully selecting the partition of $n-1$ and the structure of the idempotents, one can significantly reduce the nilpotence index of the remainder matrix $N$ compared to the trivial bound of $n$. The results provide a complete characterization for nonderogatory matrices under the specified trace and characteristic constraints, offering sharper bounds than previously available for general matrices in these settings. The paper does not claim to solve the problem for all matrices (only nonderogatory ones) or for all field characteristics without the $n \le p \le mn-1$ constraint, maintaining a modest scope focused on the structural properties of companion matrices.