As the question title suggests, what is the general classification of degree $d$ homogeneous polynomials in $\mathbb{P}^2(\text{finite field})$, up to projective equivalence?



General classification of degree $d$ homogeneous polynomials in $\mathbb{P}^2(\text{finite field})$?
Even over an algebraically closed field, there are some subtleties here. So I assume you're asking about the following action. Let $k$ be an algebraically closed field, and let $P\_d$ be the space of nonzero homogeneous polynomials of degree $d$ in $k[X,Y,Z]$, modulo the action of $\text{GL}\_3(k)$ given by $f\circ L(X,Y,Z)\sim f(X,Y,Z)$. (Or $P\_d$ could be polymomials modulo $k^\*$ and one could act via $\text{PGL\_3}(k)$ or $\text{SL}\_3(k)$.) It turns out that the quotient $P\_d/\text{GL}\_3$ is not very nice. Instead one looks at the semi-stable or stable part of $P\_d$, and then there are nice quotients. This is all explained in *Geometric Invariant Theory* (Mumford et al). He works out the 2-variable case ($\mathbb P^1$) in great detail. I don't know a reference for the 3-variable case ($\mathbb P^2$) offhand. Then, once you understand your quotient space over $\overline{\mathbb F}\_p$, you can start to study its $\mathbb F\_q$ points for finite fields, although there's still the question of whether you are looking at $\text{GL}\_3(\overline{\mathbb F\_p)}$ equivalence or $\text{GL}\_3(\mathbb F\_q)$ equivalence. 


