Determination of Some Types of Permutations over $\mathbb{F}_q^2$ with Low-Degree
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911088017145856 |
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| author | Pang, Xuan Li, Yangcheng Yuan, Pingzhi Zeng, Yuanpeng |
| author_facet | Pang, Xuan Li, Yangcheng Yuan, Pingzhi Zeng, Yuanpeng |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_01143 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Determination of Some Types of Permutations over $\mathbb{F}_q^2$ with Low-Degree Pang, Xuan Li, Yangcheng Yuan, Pingzhi Zeng, Yuanpeng Number Theory Combinatorics 11C08, 11E16, 11T06, 12E20 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. |
| title | Determination of Some Types of Permutations over $\mathbb{F}_q^2$ with Low-Degree |
| topic | Number Theory Combinatorics 11C08, 11E16, 11T06, 12E20 |
| url | https://arxiv.org/abs/2508.01143 |