Add .h_star_vector to compact rational polytopes

[jplab]

Since #25091 is merged, we can compute a few more things with normaliz. One important object is the so-called h^*-vector, see https://en.wikipedia.org/wiki/Ehrhart_polynomial#Ehrhart_series for details.

This ticket implements this method. Currently, one has to do the following:

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

which would now become:

sage: C.h_star_vector()

and return an error if the backend is not 'normaliz'...

See the related SageAsk question: https://ask.sagemath.org/question/32505/can-sage-compute-the-h-vector-of-a-polytope/

Eventually, it would be nice to also make this possible for polymake too (it would be interesting to know if they use 'normaliz' for that).

Depends on #28247

CC: @sophiasage @fchapoton @w-bruns @videlec @mkoeppe

Component: geometry

Keywords: polytopes, normaliz

Author: Sophia Elia

Branch/Commit: 48d2a6f

Reviewer: Jean-Philippe Labbé

Issue created by migration from https://trac.sagemath.org/ticket/28413

[w-bruns]

comment:1

For the h^*-vector o be well-defined, the polytope should have integral vertices. Otherwise there is no canonical choice for the denominator and, hence, the numerator of the Ehrhart series (as a rational function).

At least one should check that the denominator is a power of 1-t.

[jplab]

comment:2

Replying to @w-bruns:

For the h^*-vector o be well-defined, the polytope should have integral vertices. Otherwise there is no canonical choice for the denominator and, hence, the numerator of the Ehrhart series (as a rational function).

At least one should check that the denominator is a power of 1-t.

Okay! great, good to know. We'll put this check into the function.

[sophiasage]

New commits:

8606b7e

first changes for h-star function

[sophiasage]

Branch: public/28413

[sophiasage]

Commit: 8606b7e

[jplab]

comment:4

A few comments:

+ - ``self`` -- A lattice polytope with ``backend`` = 'normaliz'.

I would put the backend comment in a NOTE environment, although I am not completely sure. It would be to be checked in the guidelines.

Just a suggestion:

- The h*-vector as a list.
+ A list whose entries give the h*-vector.

• simplicies -> simplices

In the examples, you are only testing one of the 4 and not the four of them.

The method _h_star_vector_normaliz(self) in base should return a TypeError (as the volume_normaliz for example) and the function in backend_normaliz should be appended with _normaliz.

I would also add a less trivial example, like the 0,1 cube and perhaps hypersimplex? This would be good to get sanity checks. Further, I would include an example that uses a different backend to show even further that it requires normaliz...

Otherwise, looks good!

[sagetrac-git]

Changed commit from 8606b7e to b9df545

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

b9df545

addressing comments

[sophiasage]

Dependencies: #28247

[jplab]

Author: Sophia Elia