AI 中文总结
本文研究有限 T-四元数环上的置换多项式，先分析 $\boldsymbol{\text{Z}}_n$ 环上的置换多项式，再扩展至 T-四元数环，聚焦线性、二次及三次情形。
AI 中文摘要
T-四元数环是交换环上四元数环的一类特殊交换幺子环。置换多项式（PP）通常定义在有限域上，如素幂次有限域 $\boldsymbol{\text{F}}_q$。本文研究有限 T-四元数环上的置换多项式，重点关注线性、二次及三次情形，先探讨 $\boldsymbol{\text{Z}}_n$ 环上的置换多项式，再将讨论扩展至 T-四元数环。
英文摘要
T-quaternion rings are a special class of commutative unital subrings of the quaternion rings over commutative rings. Permutation Polynomials (PP) are typically defined over finite fields, such as $\mathbb{F}_{q}$, where $q$ is a prime power. This article examines permutation polynomials over finite T-quaternion rings, focusing specifically on linear, quadratic, and cubic cases. The study begins with permutation polynomials over the rings $\mathbb{Z}_{n}$, before extending the discussion to T-quaternion rings.
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