Counting tame $S{L}_3$- and $S{L}_4$- frieze patterns over finite fields

In this article we count tame $S{L}_3$- and $S{L}_4$-frieze patterns with width $w$ over a finite field $K$, as well as some tame $S{L}_k$-frieze patterns for higher $k$. Let $n=w+k+1$. We consider the sets $C_k(n)$ of tuples of $n$ points in the projective space ${\mathbb{P}}^{k-1}(K)$, such that $k$ consecutive points are always independent (the first and last point in the tuple are considered to be consecutive). First assume $\gcd (k,n)=1$. In this case, we prove that the problem of counting tame $S{L}_k$-frieze patterns can be reduced to counting $C_k(n)$. We also show that $\mid C_k(n)\mid$ is essentially already known as long as $k$ and $n$ are coprime, and we derive the number of tame $S{L}_k$-frieze-patterns in that case. In the case $\gcd (k,n)\mathrm{\ne }1$, we define certain subsets $C_k^{\ast }(n)$ and show that it is sufficient to count these sets. Afterwards, we count $C_k^{\ast }(n)$ in the cases $k=3$ and $k=4$ and thus the number of tame $S{L}_3$- and $S{L}_4$-frieze patterns for any width $w$.