  Given a set $\mathcal A = \{a_1,\ldots,a_n\} \subset \mathbb{N}^m$ of nonzero
vectors defining a simplicial toric ideal $I_{\mathcal A} \subset
k[x_1,...,x_n]$, where $k$ is an arbitrary field, we provide an algorithm for
checking whether $I_{\mathcal A}$ is a complete intersection. This algorithm
does not require the explicit computation of a minimal set of generators of
$I_{\mathcal A}$. The algorithm is based on the application of some new results
concerning toric ideals to the simplicial case. For homogenous simplicial toric
ideals, we provide a simpler version of this algorithm. Moreover, when $k$ is
an algebraically closed field, we list all ideal-theoretic complete
intersection simplicial projective toric varieties that are either smooth or
have one singular point.
