Howard Nelson

2022-11-22

Affine curve of an absolutely irreducible polynomial
Let $f\in {\mathbb{F}}_{q}\left[X,Y\right]$ be an absolutely irreducible polynomial of degree n. Denote the affine curve defined by the equation $f\left(X,Y\right)=0$ over ${\mathbb{F}}_{q}$ by $\mathrm{\Gamma }\left({\mathbb{F}}_{q}\right)$, and let $d=\mathrm{deg}\mathrm{\Gamma }$. The for each m there exists ${q}_{0}$ such that $|\mathrm{\Gamma }\left({\mathbb{F}}_{q}\right)|\ge m$ for all $q\ge {q}_{0}$.

Kaeden Lara

Expert

Step 1
It is trivial as we have the inequality $|\mathrm{\Gamma }\left({\mathbb{F}}_{q}\right)|\ge q+1+\left(d-1\right)\left(d-2\right)\sqrt{q}-d$Step 2
Now $q+1$ grows faster than $\left(d-1\right)\left(d-2\right)\sqrt{q}-d$

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