Complete intersections in simplicial toric varieties

Given a set A={a1,…,an}⊂NmA={a1,…,an}⊂Nm of nonzero vectors defining a simplicial toric ideal IA⊂k[x1,…,xn]IA⊂k[x1,…,xn], where k   is an arbitrary field, we provide an algorithm for checking whether IAIA is a complete intersection. This algorithm does not require the explicit computation of a minimal set of generators of IAIA. The algorithm is based on the application of some new results concerning toric ideals to the simplicial case. For homogeneous 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.