Application of the study of equations over finite fields

Question

I’m currently studying equations over finite fields (in particular solutions over $F_{q^k}$ to equations of the form $y^q-y=f(X)$ for a polynomial $f(X) \in F_q[X]$) and approximations of the number of their solutions.
In the above case (with N being the number of solutions $(x,y) \in (F_{q^k})^2$ to $y^q-y=f(X)$) we know that $|N-q^k|<q^{[\frac{k}{2}]+4}$.

I am wondering now, how those results, that we find, can be applied (e.g. in computer science). Why are we even studying those equations? Can someone provide me with an example of an application?

Thank you a lot in advance!

Answer 1

Question: "I am wondering now, how those results, that we find, can be applied (e.g. in computer science)."

Answer: Studying algebraic curves over finite fields has applications in the field "cryptography":

https://en.wikipedia.org/wiki/Elliptic-curve_cryptography

Example: In Hartshorne (HH), Exercise AppC.5.6 and 5.7 they prove the Weil conjectures for elliptic curves. Section IV in the same book studies curves in general. Fulton's "Algebraic curves" is a more elementary introduction to the theory of algebraic curves. Note that in positive characteristic you will need much of the machinery introduced in HH.

In algebra/geometry the study of algebraic varieties over a finite field is much studied. This study goes all the way back to 1800BC.

https://en.wikipedia.org/wiki/Weil_conjectures

https://en.wikipedia.org/wiki/Algebraic_number_theory#Diophantus

Answer 2

There is a class of error-correcting codes called "algebraic geometry codes". The parameters for these codes depend on the number of solutions of equations over finite fields. https://arxiv.org/pdf/1505.03020.pdf claims to be an introductory paper on the topic.