Let L-1,..., L-s be line bundles on a smooth complex variety X subset of P-r and let D-1,..., D-s be divisors on X such that D-i represents L-i. We give a probabilistic algorithm for computing the degree of intersections of polar classes which are in turn used for computing the Euler characteristic of linear combinations of L-1,..., L-s. The input consists of generators for the homogeneous ideals I-X, I-Di subset of C[x(0),..., x(r).] defining X and D-i.