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