On the Length of Generating Sets with Conditions on Minimal Polynomial

doi:10.1007/s42967-025-00507-0

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (4): 1427-1449.doi: 10.1007/s42967-025-00507-0

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On the Length of Generating Sets with Conditions on Minimal Polynomial

Chengjie Wang

College of Sciences, Shanghai University, Shanghai, 200444, China

Received:2025-03-26 Revised:2025-05-21 Accepted:2025-05-23 Online:2026-08-20 Published:2026-09-17
Supported by:
I would like to express my sincere gratitude to Shitov for sharing their latest results with me via email. These findings have made significant contributions to this research field.

Abstract

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 n/2, where n denotes the order of the matrix. Compared to the bound provided in Theorem 3.1 of Guterman et al. (Linear Algebra Appl 543: 234–250, 2018), this result reduces the constraints on the Jordan canonical forms. In addition, it is demonstrated that the bound 7n/2-4 holds when the generating set contains a matrix with a minimal polynomial of degree t satisfying 2t ≤ n ≤ 3t-1. The primary enhancements consist of quantitative bounds and reduced reliance on Jordan form structural constraints.

Key words: The full matrix algebra, Length of an algebra, Generating systems, The degree of minimal polynomial

Chengjie Wang. On the Length of Generating Sets with Conditions on Minimal Polynomial[J]. Communications on Applied Mathematics and Computation, 2026, 8(4): 1427-1449.

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On the Length of Generating Sets with Conditions on Minimal Polynomial

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