Chevalley's theorem says that if $k$ is a finite field and $f(X_1,...,X_n)$ is a form (homogeneous polynomial) of degree $d > n$, then the equation $f(X_1,...,X_n) = 0$ has a non-trivial solution in $k^n$.

I am interested in examples showing that this is optimal, i.e. forms $f(X_1,...,X_n)$ of degree $d = n$ which have only the trivial zero. I call such a form anisotropic (is this "official" terminology?). What are the known examples of such forms? Can we classify/describe them? I am in particular interested in the case where $n = d = 4$, but any relevant information about the other cases is also more than welcome.