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Let $f(x,y)$ be an affine curve over a finite field $\mathbb{F}_q$. Assume that it has some rational points over $\mathbb{F}_{q^r}$, i.e. it has some $\mathbb{F}_{q^r}$-rational points for some integer $r$. Is there any method to check which of these $\mathbb{F}_{q^r}$-rational points are lying on $\mathbb{F}_q$?

In other words, let's say $f(x,y)$ has n $\mathbb{F}_{q^r}$-rational points, how many of these $n$ points can be lying on $\mathbb{F}_q$?

Thank you very much in advance!

nomadd
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    Not much can be said in general. Too little data here. There are examples of situations where extending the field won't give any new points (and all $n$ points are in the base field). Weil "conjectures" (in the case of curves theorems he proved himself, and in general theorems proved by Deligne) tell a lot. If we denote by $N_r$ the number of points over $\Bbb{F}_{q^r}$, and the curve is smooth, then knowing $N_1,N_2,\ldots,N_g$ gives the rest. So the genus of the curve $g$ plays a role. – Jyrki Lahtonen Jan 02 '22 at 06:20
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    (cont'd) The reverse direction of going from $N_r$ to $N_1$ is tricky even in the $g=1$ case, because the process involves taking complex roots of zeros of the zeta function, and those are not unique. – Jyrki Lahtonen Jan 02 '22 at 06:21
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    A theoretical tool is the observation that a point $P=(x,y)$ with coordinates in $\overline{\Bbb{F}_q}$ has coordinates in the subfield $\Bbb{F}_q$ if and only if $P$ is a fixed point of the appropriate Frobenius mapping $F:(x,y)\mapsto (x^q,y^q)$. This does help in building the theory, in particular in the case of elliptic curves. But it isn't very useful for the task at hand, if all we are given is the polynomial $f(x,y)$. – Jyrki Lahtonen Jan 02 '22 at 06:26

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