Ehrhart series coefficients and quasi-period for random rational polytopes

Ehrhart series coefficients and quasi-period for random rational polytopes

A dataset of Ehrhart data for 84000 randomly generated rational polytopes, in dimensions 2 to 4, with quasi-periods 2 to 15.

The polytopes used to generate this data were produced by the following algorithm:

1. Fix \(d\) a positive integer in \(\{2,3,4\}\).
2. Choose \(r\in\{2,\ldots,15\}\) uniformly at random.
3. Choose \(d + k\) lattice points \(\{v_1,\ldots,v_{d+k}\}\) uniformly at random in a box \([-5r,5r]^d\), where \(k\) is chosen uniformly at random in \(\{1,\ldots,5\}\).
4. Set \(P := \mathrm{conv}\{v_1,\ldots,v_{d+k}\}\). If \(\mathrm{dim}(P)\) is not equal to \(d\) then return to step 3.
5. Choose a lattice point \(v\in P \cap \mathbb{Z}^d\) uniformly at random and replace \(P\) with the translation \(P-v\).
6. Replace \(P\) with the dilation \(P/r\).

The final dataset was produced by first removing duplicate records, and then downsampling to a subset with 2000 datapoints for each pair \((d,q)\), where \(d\) is the dimension of \(P\) and \(q\) is the quasi-period of \(P\), with \(d\in\{2,3,4\}\) and \(q\in\{2,\ldots,15\}\).

For details, see the paper:

 Machine Learning the Dimension of a Polytope, Tom Coates, Johannes Hofscheier, and Alexander M. Kasprzyk, 2022.

If you make use of this data, please cite the above paper and the DOI for this data:

 doi:10.5281/zenodo.6614829

quasiperiod.txt.gz
The file "quasiperiod.txt.gz" is a gzip-compressed plain text file containing key:value records with keys and values as described below, where each record is separated by a blank line. There are 84000 records in the file.

Example record
ULID: 01G57JBYP2ZW825E0NT4Q9JQNQ
Dimension: 2
Quasiperiod: 2
Volume: 97
EhrhartDelta: [1,50,195,289,192,49]
Ehrhart: [1,50,198,...]
LogEhrhart: [0.000000000000000000000000000000,3.91202300542814605861875078791,5.28826703069453523626966617327,...]

(The values for Ehrhart and LogEhrhart in the example have been truncated.)

For each polytope \(P\) of dimension \(d\) and quasi-period \(q\) we record the following keys and values in the dataset:

ULID: A randomly generated string identifier for this record.
Dimension: A positive integer. The dimension \(2 \leq d \leq 4\) of the polytope \(P\).
Quasiperiod: A positive integer. The quasi-period \(2 \leq q \leq 15\) of the polytope \(P\).
Volume: A positive rational number. The lattice-normalised volume \(\mathrm{Vol}(P)\) of the polytope \(P\).
EhrhartDelta: A sequence \([1,a_1,a_2,\ldots,a_N]\) of integers of length \(N + 1\), where \(N := q(d + 1) - 1\). This is the Ehrhart \(\delta\)-vector (or \(h^*\)-vector) of \(P\). The Ehrhart series \(\mathrm{Ehr}(P)\) of \(P\) is given by the power-series expansion of \((1 + a_1t + a_2t^2 + \ldots + a_Nt^N) / (1 - t^q)^{d+1}\).
Ehrhart: A sequence \([1,c_1,c_2,\ldots,c_{1100}]\) of positive integers. The value \(c_i\) is equal to the number of lattice points in the \(i\)-th dilation of \(P\), that is, \(c_i = \#(iP \cap \mathbb{Z}^d)\). Equivalently, \(c_i\) is the coefficient of \(t^i\) in \(\mathrm{Ehr}(P) = 1 + c_1t + c_2t^2 + \ldots = (1 + a_1t + a_2t^2 + \ldots + a_Nt^N) / (1 - t^q)^{d+1}\).
LogEhrhart: A sequence \([0,y_1,y_2,\ldots,y_{1100}]\) of non-negative floating point numbers. Here \(y_i := \log c_i\).