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

On the Length of Generating Sets with Conditions on Minimal Polynomial

Chengjie Wang

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

收稿日期:2025-03-26 修回日期:2025-05-21 接受日期:2025-05-23 出版日期:2026-08-20 发布日期:2026-09-17
通讯作者: Chengjie Wang, Email: wangchengjie@shu.edu.cn E-mail:wangchengjie@shu.edu.cn
基金资助:
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.

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.

摘要/Abstract

摘要： 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.

关键词: The full matrix algebra, Length of an algebra, Generating systems, The degree of minimal polynomial

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

On the Length of Generating Sets with Conditions on Minimal Polynomial

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