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I want to investigate rational dilates of rational polytopes. I am aware that the Ehrhart-Polynomial of such polytopes is a quasipolynomial, i.e. the coefficients are periodic functions c_i(lambda), depending on the (rational) scaling factor lambda. Below the output for the function EHRHART_QUASI_POLYNOMIAL called on the interval P = [1.5,2.5] scaled by 1/15.

For this specific scaling factor lambda I was expecting one single polynomial, where the coefficients are the periodic function values c_i(1/15).
However it seems polymake gives me somehow the whole periodic function. There are 30 (=2*15=2*codenominator of P) polynomials printed out, hinting towards the known rational Ehrhart theory. Unfortunately I can't make much more sense of what I'm looking at. Some more insight on what the function is doing and how to interpret the output would be very much appreciated.

Thanks in advance, Richard

In[]: $p_ex0 = new Polytope<Rational>(VERTICES=>[[1,3/2],[1,5/2]]);
$p_ex1 = scale($p_ex0,1/15);
print join("\n",@{$p_ex1>EHRHART_QUASI_POLYNOMIAL});

Out[]: 1/15*x + 1
1/15*x -1/15
1/15*x -2/15
1/15*x -1/5
1/15*x -4/15
1/15*x -1/3
1/15*x + 3/5
1/15*x + 8/15
1/15*x + 7/15
1/15*x + 2/5
1/15*x + 1/3
1/15*x -11/15
1/15*x + 1/5
1/15*x + 2/15
1/15*x + 1/15
1/15*x
1/15*x -1/15
1/15*x -2/15
1/15*x + 4/5
1/15*x + 11/15
1/15*x + 2/3
1/15*x -2/5
1/15*x -7/15
1/15*x -8/15
1/15*x + 2/5
1/15*x + 1/3
1/15*x + 4/15
1/15*x + 1/5
1/15*x + 2/15
1/15*x + 1/15

A quasi-polynomial can be written in two ways, either as a polynomial with periodic functions as coefficients, or, as a list of polynomials for each modulus up to the quasi-period n, i.e.,
$$ f(k) = f_i(k) \quad \text{if} \quad k \equiv i \ (\text{mod}\ n) \ .$$
The polynomials in this property are these 30 polynomials \( f_i \), as returned by our interface to normaliz, see also: https://github.com/Normaliz/Normaliz/bl ... rmaliz.pdf

Best,
Benjamin

Thank you, Benjamin, for the explanation. This helps a lot.
I still need to look at some more examples to get a feel for the quasipolynomials and their counting. Im guessing that the periodicity somehow conserves the fact that we have 'large' areas without adding new lattice points to \lambda*P, while scaling by a rational \lambda, i.e. we still count finitely many integer points. But thats just a guess for now