Can sage compute the h* vector of a polytope?

polytope

asked 2016-02-09 07:46:30 +0200

done_with_fish
273 ●6 ●16 ●32

updated 2016-02-09 19:36:20 +0200

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?

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

tmonteil ( 2016-02-09 15:53:44 +0200 )edit

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

kcrisman ( 2016-02-09 18:06:34 +0200 )edit

@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!

done_with_fish ( 2016-02-09 19:28:24 +0200 )edit

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

kcrisman ( 2016-02-09 23:05:25 +0200 )edit

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

done_with_fish ( 2016-02-10 19:14:30 +0200 )edit

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answered 2018-05-02 15:24:44 +0200

jipilab
533 ●17 ●18

updated 2019-08-27 16:21:01 +0200

You can use the normaliz backend (requires Normaliz 3.5.4) and its python interface pynormaliz (requires PyNormaliz 1.16).

You can install them by typing:

sage -i normaliz
sage -i pynormaliz

Then, in a terminal with Sage 8.9 or more recent, you can get the h^*-vector by typing:

sage: C = polytopes.hypercube(3, backend="normaliz")
sage: C.ehrhart_series().numerator().coefficients()
[1, 23, 23, 1]

This hypercube is the ±1 cube, so its volume is 8*factorial(3)=48, which is 1+23+23+1.

Eventually, once this ticket is merged, it will be possible to call it directly on the polytope like so:

sage: C.h_star_vector()
[1, 23, 23, 1]

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answered 2016-02-17 11:04:44 +0200

kdilks
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updated 2016-02-17 19:14:12 +0200

kcrisman
12222 ●41 ●135 ●254

Sage doesn't have a direct 'print out the h* vector' command. It's not too difficult to manipulate the Ehrhart polynomial to get it, though I'm not sure my way could be cleanly implemented into Sage proper.

Let's say P is your polytope, of dimension n=P.dimension(), with Ehrhart polynomial p=P.ehrhart_polynomial() . Also, let t=p.parent().gen() . We need a polynomial ring to work over, and this let's us work over the same polynomial ring as p. Now, compute the first n+2 terms of the Ehrhart series (or more, if you want to be safe)

X=0
for i in range(0,n+10):
    X+=p(i)*t^i
print(X)

Now, look at X*(1-t)^(n+1) . If X were the entire infinite Ehrhart series, we would get exactly the h* vector. But since we took sufficiently many terms of the Ehrhart series, we'll get lower order terms corresponding to the h* vector, and higher order error terms that can be discarded.

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Can you add a specific example of this for a slightly nontrivial polytope?

kcrisman ( 2016-02-17 19:13:38 +0200 )edit

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