# Can sage compute the h* vector of a polytope?

We can define polytopes in sage with the command

```
sage: p = Polytope(A)
```

Here p is the convex hull of the integer matrix A.

I'm curious if we can compute the h* vector of p. Is this possible?

Roughly, the h* vector of a convex lattice polytope is constructed as follows.

The Ehrhart series can be expressed as a rational function whose numerator is a polynomial. The h* vector of a polytope is the vector of coefficients of this polynomial.

I'm aware that sage can compute the ehrhart polynomial of a polytope with `sage: p.ehrhart_polynomial()`

but I cant find anything about the h* vector in the documentation.

I know the command in polymake is `$p -> H_STAR_VECTOR`

, but I'm not sure if this vector can be constructed in sage.

Perhaps a more general question would be: can I pass a polytope defined in sage to polymake and can sage read the results of the polymake operation?

If you could please give us a pointer to the definition of the h* vector of a polytope ?

@done_with_fish: Does the ref for h-rep and v-rep of polytopes help?

@tmonteil I've included a brief description as well as a link to the wikipedia article where the h* vector is defined. Thanks for looking at this!

http://trac.sagemath.org/ticket/14116 is probably relevant here...

@kcrisman If I'm reading that link correctly, it sounds like sage once had this capability but no longer does?