Permutation Rational Functions over Quadratic Extensions of Finite Fields
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916104135245824 |
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| author | Chen, Ruikai Mesnager, Sihem |
| author_facet | Chen, Ruikai Mesnager, Sihem |
| contents | Permutation rational functions over finite fields have attracted much attention in recent years. In this paper, we introduce a class of permutation rational functions over $\mathbb F_{q^2}$, whose numerators are so-called $q$-quadratic polynomials. To this end, we will first determine the exact number of zeros of a special $q$-quadratic polynomial in $\mathbb F_{q^2}$, by calculating character sums related to quadratic forms of $\mathbb F_{q^2}/\mathbb F_q$. Then given some rational function, we can demonstrate whether it induces a permutation of $\mathbb F_{q^2}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_04121 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Permutation Rational Functions over Quadratic Extensions of Finite Fields Chen, Ruikai Mesnager, Sihem Number Theory 11T06, 12E10, 51E15, 11R58, 11T06, 11T55, 14H05 Permutation rational functions over finite fields have attracted much attention in recent years. In this paper, we introduce a class of permutation rational functions over $\mathbb F_{q^2}$, whose numerators are so-called $q$-quadratic polynomials. To this end, we will first determine the exact number of zeros of a special $q$-quadratic polynomial in $\mathbb F_{q^2}$, by calculating character sums related to quadratic forms of $\mathbb F_{q^2}/\mathbb F_q$. Then given some rational function, we can demonstrate whether it induces a permutation of $\mathbb F_{q^2}$. |
| title | Permutation Rational Functions over Quadratic Extensions of Finite Fields |
| topic | Number Theory 11T06, 12E10, 51E15, 11R58, 11T06, 11T55, 14H05 |
| url | https://arxiv.org/abs/2309.04121 |