Permutation polynomials, projective polynomials, and bijections between $μ_{\frac{q^n-1}{q-1}}$ and $PG(n-1,q)$
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| Format: | Preprint |
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2025
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| _version_ | 1866913867068604416 |
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| author | Lin, Tong Wang, Qiang |
| author_facet | Lin, Tong Wang, Qiang |
| contents | Using arbitrary bases for the finite field $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$, we obtain the generalized Möbius transformations (GMTs), which are a class of bijections between the projective geometry $PG(n-1,q)$ and the set of roots of unity $μ_{\frac{q^n-1}{q-1}}\subseteq\mathbb{F}_{q^n}$, where $n\geq 2$ is any integer. We also introduce a class of projective polynomials, using the properties of which we determine the inverses of the GMTs. Moreover, we study the roots of those projective polynomials, which lead to a three-way correspondence between partitions of $\mathbb{F}_{q^n}^\ast,μ_{\frac{q^n-1}{q-1}}$ and $PG(n-1,q)$. Through this correspondence and the GMTs, we construct permutation polynomials of index $\frac{q^n-1}{q-1}$ over $\mathbb{F}_{q^n}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_11775 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Permutation polynomials, projective polynomials, and bijections between $μ_{\frac{q^n-1}{q-1}}$ and $PG(n-1,q)$ Lin, Tong Wang, Qiang Combinatorics Number Theory 11T06 Using arbitrary bases for the finite field $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$, we obtain the generalized Möbius transformations (GMTs), which are a class of bijections between the projective geometry $PG(n-1,q)$ and the set of roots of unity $μ_{\frac{q^n-1}{q-1}}\subseteq\mathbb{F}_{q^n}$, where $n\geq 2$ is any integer. We also introduce a class of projective polynomials, using the properties of which we determine the inverses of the GMTs. Moreover, we study the roots of those projective polynomials, which lead to a three-way correspondence between partitions of $\mathbb{F}_{q^n}^\ast,μ_{\frac{q^n-1}{q-1}}$ and $PG(n-1,q)$. Through this correspondence and the GMTs, we construct permutation polynomials of index $\frac{q^n-1}{q-1}$ over $\mathbb{F}_{q^n}$. |
| title | Permutation polynomials, projective polynomials, and bijections between $μ_{\frac{q^n-1}{q-1}}$ and $PG(n-1,q)$ |
| topic | Combinatorics Number Theory 11T06 |
| url | https://arxiv.org/abs/2501.11775 |