$\mathbb{F}_q$-primitive points on varieties over finite fields
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arXiv
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| Format: | Preprint |
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2024
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| author | Takshak, Soniya Kapetanakis, Giorgos Sharma, Rajendra Kumar |
| author_facet | Takshak, Soniya Kapetanakis, Giorgos Sharma, Rajendra Kumar |
| contents | Let $r$ be a positive divisor of $q-1$ and $f(x,y)$ a rational function of degree sum $d$ over $\mathbb{F}_q$ with some restrictions, where the degree sum of a rational function $f(x,y) = f_1(x,y)/f_2(x,y)$ is the sum of the degrees of $f_1(x,y)$ and $f_2(x,y)$. In this article, we discuss the existence of triples $(α, β, f(α, β))$ over $\mathbb{F}_q$, where $α, β$ are primitive and $f(α, β)$ is an $r$-primitive element of $\mathbb{F}_q$. In particular, this implies the existence of $\mathbb{F}_q$-primitive points on the surfaces of the form $z^r = f(x,y)$. As an example, we apply our results on the unit sphere over $\mathbb{F}_q$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_03836 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\mathbb{F}_q$-primitive points on varieties over finite fields Takshak, Soniya Kapetanakis, Giorgos Sharma, Rajendra Kumar Number Theory 12E20, 11T23 Let $r$ be a positive divisor of $q-1$ and $f(x,y)$ a rational function of degree sum $d$ over $\mathbb{F}_q$ with some restrictions, where the degree sum of a rational function $f(x,y) = f_1(x,y)/f_2(x,y)$ is the sum of the degrees of $f_1(x,y)$ and $f_2(x,y)$. In this article, we discuss the existence of triples $(α, β, f(α, β))$ over $\mathbb{F}_q$, where $α, β$ are primitive and $f(α, β)$ is an $r$-primitive element of $\mathbb{F}_q$. In particular, this implies the existence of $\mathbb{F}_q$-primitive points on the surfaces of the form $z^r = f(x,y)$. As an example, we apply our results on the unit sphere over $\mathbb{F}_q$. |
| title | $\mathbb{F}_q$-primitive points on varieties over finite fields |
| topic | Number Theory 12E20, 11T23 |
| url | https://arxiv.org/abs/2410.03836 |