  Let $n$ be a positive integer, and let $R$ be a (possibly infinite
dimensional) finitely presented algebra over a computable field of
characteristic zero. We describe an algorithm for deciding (in principle)
whether $R$ has at most finitely many equivalence classes of $n$-dimensional
irreducible representations. When $R$ does have only finitely many such
equivalence classes, they can be effectively counted (assuming that $k[x]$
posesses a factoring algorithm).
