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arXiv · 2508.01143

Determination of Some Types of Permutations over $\mathbb{F}_q^2$ with Low-Degree

Xuan Pang·Yangcheng Li·Pingzhi Yuan·Yuanpeng Zeng
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Abstract

The characterization of permutations over finite fields is an important topic in number theory with a long-standing history. This paper presents a systematic investigation of low-degree bivariate polynomial systems $F=(f_1(x,y),f_2(x,y))$ defined over $\mathbb{F}_{q}^2$. Specifically, we employ Hermite's Criterion to completely classify bivariate quadratic permutation polynomial systems, while utilizing the theory of permutation rational functions to give a full classification of bivariate 3-homogeneous permutation polynomial systems. Furthermore, as an application of our findings, we provide an explicit characterization of the permutation binomials of the form $x^3+ax^{2q+1}$ over $\mathbb{F}_{q^2}$ with characteristic $p\neq3$, thereby resolving a significant special case within this classical research domain.

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Xuan Pang, Yangcheng Li, Pingzhi Yuan, Yuanpeng Zeng. 2025-08-02. Determination of Some Types of Permutations over $\mathbb{F}_q^2$ with Low-Degree. https://arxiv.org/abs/2508.01143

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