Hello,
First, sorry for the delay. As far as I know, the notion of \( h^* \)-vectors for non-integral polytopes is not really well established as there is no canonical choice for the denominator of the Hilbert series. In hindsight we probably should have restricted this property to lattice polytopes.
In the non-integral case it is best to work with directly the property HILBERT_SERIES.
Our choice is derived from our interface to libnormaliz, more precisely, it is the first variant of the numerator of the Hilbert series as mentioned in Section 2.5 of the
normaliz manual.
To compute the coefficients according to the wikipedia page you could do the following (continuing with your example):
Code: Select all
$den = lcm(primitive($p->VERTICES)->col(0));
$hs = new RationalFunction($p->HILBERT_SERIES);
$hs *= 1-new UniMonomial(new Int($den)) foreach (1..$p->CONE_DIM);
print numerator($hs);
3*x^4 + 9*x^3 + 10*x^2 + 5*x + 1
Best,
Benjamin
PS: We will also make some improvements for the property H_STAR_VECTOR (at least concerning the documentation) for the next major release.