Let $X$ be a smooth projective toric variety. Do any of the math computer algebra systems have an algorithm implemented to compute the Hodge numbers of a generic complete intersection in $X$? Say in terms of the combinatorics used to encode divisors.
Even more basic, how about for hypersurfaces in a given ample class? i.e. given an integral polytope, it would output the Hodge polynomial.
If not, why not? Is an algorithm too difficult to implement? Or are there still unknowns involved, e.g. ranks of various maps of cohomology spaces in long exact sequences?
