On the length of generating sets with conditions on minimal polynomial
Chengjie Wang
TL;DR
The paper investigates Paz's conjecture on the linear upper bound for the length $\ell(\mathcal{S})$ of generating systems of the full matrix algebra $M_n(\mathbb{F})$, focusing on how algebraic invariants like the degree of the minimal polynomial $m(\mathcal{S})$ influence the bound. It develops a framework where invariance under invertible transformations and shifts, together with rank-based invariants, yield explicit linear bounds, notably $\ell(\mathcal{S})\le 3n-5$ when $m(\mathcal{S})>\tfrac{n}{2}$ and $\ell(\mathcal{S})\le \tfrac{7n}{2}-4$ when $2t\le n\le 3t-1$ with $m(\mathcal{S})=t$. The results reduce reliance on detailed Jordan-form configurations by leveraging spectral data to bound generation length, and they confirm linear bounds under broader conditions (e.g., nonderogatory matrices) that align with Paz-type goals. Overall, the work clarifies how invariants like the minimal polynomial degree and rank within generating sets constrain the length, providing sharper, condition-driven linear bounds with potential implications for related algebras and group algebras.
Abstract
Linear upper bounds may be derived by imposing specific structural conditions on a generating set, such as additional constraints on ranks, eigenvalues, or the degree of the minimal polynomial of the generating matrices. This paper establishes a linear upper bound of \(3n-5\) for generating sets that contain a matrix whose minimal polynomial has a degree exceeding \(\frac{n}{2}\), where \(n\) denotes the order of the matrix. Compared to the bound provided in \cite[Theorem 3.1]{r2}, this result reduces the constraints on the Jordan canonical forms. Additionally, it is demonstrated that the bound \(\frac{7n}{2}-4\) holds when the generating set contains a matrix with a minimal polynomial of degree \(t\) satisfying \(2t\le n\le 3t-1\). The primary enhancements consist of quantitative bounds and reduced reliance on Jordan form structural constraints.