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,...,15} uniformly at random.
 3) Choose d + k lattice points {v_1,...,v_(d+k)} uniformly at random in a box [-5r,5r]^d, where k is chosen uniformly at random in {1,...,5}.
 4) Set P := conv{v_1,...,v_(d+k)}. If dim(P) is not equal to d then return to step 3.
 5) Choose a lattice point v in P \cap 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,...,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 <= d <= 4 of the polytope P.
Quasiperiod: A positive integer. The quasi-period 2 <= q <= 15 of the polytope P.
Volume: A positive rational number. The lattice-normalised volume Vol(P) of the polytope P.
EhrhartDelta: A sequence [1,a_1,a_2,...,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 Ehr(P) of P is given by the power-series expansion of (1 + a_1*t + a_2*t^2 + ... + a_N*t^N) / (1 - t^q)^(d+1).
Ehrhart: A sequence [1,c_1,c_2,...,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 Z^d). Equivalently, c_i is the coefficient of t^i in Ehr(P) = 1 + c_1*t + c_2*t^2 + ...
LogEhrhart: A sequence [0,y_1,y_2,...,y_1100] of non-negative floating point numbers. Here y_i := log c_i.