  Let $n$ be a positive integer, and let $k$ be a field (of arbitrary
characteristic) accessible to symbolic computation. We describe an algorithmic
test for determining whether or not a finitely presented $k$-algebra $R$ has
infinitely many equivalence classes of semisimple representations $R \to
M_n(k')$, where $k'$ is the algebraic closure of $k$. The test reduces the
problem to computational commutative algebra over $k$, via famous results of
Artin, Procesi, and Shirshov. The test is illustrated by explicit examples,
with $n = 3$.
