$r$-primitive $k$-normal polynomials over finite fields with last two coefficients prescribed
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915244849233920 |
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| author | Chatterjee, K. Sharma, R. K. Tiwari, S. K. |
| author_facet | Chatterjee, K. Sharma, R. K. Tiwari, S. K. |
| contents | Let $ξ\in\mathbb{F}_{q^m}$ be an $r$-primitive $k$-normal element over $\mathbb{F}_q$, where $q$ is a prime power and $m$ is a positive integer. The minimal polynomial of $ξ$ is referred to be the $r$-primitive $k$-normal polynomial of $ξ$ over $\mathbb{F}_q$. In this article, we study the existence of an $r$-primitive $k$-normal polynomial over $\mathbb{F}_q$ such that the last two coefficients are prescribed. In this context, first, we prove a sufficient condition which guarantees the existence of such a polynomial. Further, we compute all possible exceptional pairs $(q,m)$ in case of $3$-primitive $1$-normal polynomials for $m\geq 7$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04999 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $r$-primitive $k$-normal polynomials over finite fields with last two coefficients prescribed Chatterjee, K. Sharma, R. K. Tiwari, S. K. Number Theory 12E20, 11T23 Let $ξ\in\mathbb{F}_{q^m}$ be an $r$-primitive $k$-normal element over $\mathbb{F}_q$, where $q$ is a prime power and $m$ is a positive integer. The minimal polynomial of $ξ$ is referred to be the $r$-primitive $k$-normal polynomial of $ξ$ over $\mathbb{F}_q$. In this article, we study the existence of an $r$-primitive $k$-normal polynomial over $\mathbb{F}_q$ such that the last two coefficients are prescribed. In this context, first, we prove a sufficient condition which guarantees the existence of such a polynomial. Further, we compute all possible exceptional pairs $(q,m)$ in case of $3$-primitive $1$-normal polynomials for $m\geq 7$. |
| title | $r$-primitive $k$-normal polynomials over finite fields with last two coefficients prescribed |
| topic | Number Theory 12E20, 11T23 |
| url | https://arxiv.org/abs/2501.04999 |