Howard Nelson

Answered

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 $q\ge {q}_{0}$.

Answer & Explanation

Kaeden Lara

Expert

2022-11-23Added 23 answers

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 $\left(d-1\right)\left(d-2\right)\sqrt{q}-d$

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?