Permutation Polynomials of the form $L(X)+γTr_q^{q^3}(h(X))$ over finite fields with even characteristic
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| Format: | Preprint |
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2025
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| _version_ | 1866917170314739712 |
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| author | Pang, Xuan Wu, Danyao Yuan, Pingzhi |
| author_facet | Pang, Xuan Wu, Danyao Yuan, Pingzhi |
| contents | Permutation polynomials over finite fields have extensive applications in various areas. Particularly, permutation polynomials with simple forms are of great interest. In recent papers, several classes of permutation polynomials of the form $L(X)+Tr_q^{q^3}(h(X))$ have been constructed. This paper further investigates permutation polynomials of such form over $\mathbb{F}_{q^3}$. Unlike previous studies, we transform the problem of constructing univariate permutation polynomials over finite fields into that of constructing corresponding multivariate permutations over $\mathbb{F}_{q}$-vector spaces. Through this approach, we completely characterize a class of permutation polynomials of the form $L(X)+γTr_q^{q^3}(c_1X+c_2X^2+c_3X^3+c_4X^{q+2})$ over $\mathbb{F}_{q^3}$, where $q=2^m$, $L(X)=X^q+aX$ and $a,c_1,c_2,c_3,c_4,γ\in\mathbb{F}_q$ with $a^2+a+1\neq0$. Furthermore, using a similar method, we generalize several results from a recent work by Jiang, Li and Qu (2026). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_21908 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Permutation Polynomials of the form $L(X)+γTr_q^{q^3}(h(X))$ over finite fields with even characteristic Pang, Xuan Wu, Danyao Yuan, Pingzhi Number Theory 11T06, 11T55 Permutation polynomials over finite fields have extensive applications in various areas. Particularly, permutation polynomials with simple forms are of great interest. In recent papers, several classes of permutation polynomials of the form $L(X)+Tr_q^{q^3}(h(X))$ have been constructed. This paper further investigates permutation polynomials of such form over $\mathbb{F}_{q^3}$. Unlike previous studies, we transform the problem of constructing univariate permutation polynomials over finite fields into that of constructing corresponding multivariate permutations over $\mathbb{F}_{q}$-vector spaces. Through this approach, we completely characterize a class of permutation polynomials of the form $L(X)+γTr_q^{q^3}(c_1X+c_2X^2+c_3X^3+c_4X^{q+2})$ over $\mathbb{F}_{q^3}$, where $q=2^m$, $L(X)=X^q+aX$ and $a,c_1,c_2,c_3,c_4,γ\in\mathbb{F}_q$ with $a^2+a+1\neq0$. Furthermore, using a similar method, we generalize several results from a recent work by Jiang, Li and Qu (2026). |
| title | Permutation Polynomials of the form $L(X)+γTr_q^{q^3}(h(X))$ over finite fields with even characteristic |
| topic | Number Theory 11T06, 11T55 |
| url | https://arxiv.org/abs/2512.21908 |