  We consider the problem of computing the Euler characteristic of an abstract
simplicial complex given by its vertices and facets. We show that this problem
is #P-complete and present two new practical algorithms for computing Euler
characteristic. The two new algorithms are derived using combinatorial
commutative algebra and we also give a second description of them that requires
no algebra. We present experiments showing that the two new algorithms can be
implemented to be faster than previous Euler characteristic implementations by
a large margin.
