Ehrhart series coefficients for random lattice polytopes

Ehrhart series coefficients for random lattice polytopes

A dataset of Ehrhart data for 2918 randomly generated lattice polytopes, in dimensions 2 to 8.

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

Fix \(d\) a positive integer in \(\{2,\ldots,8\}\).

Choose \(d + k\) lattice points \(\{v_1,\ldots,v_{d+k}\}\) uniformly at random in a box \([-5,5]^d\), where \(k\) is chosen uniformly at random in \(\{1,\ldots,5\}\).

Set \(P := \mathrm{conv}\{v_1,\ldots,v_{d+k}\}\). If \(\mathrm{dim}(P)\neq d\) then return to step 2.

The final dataset has duplicate records removed. The data is distributed by dimension \(d\) as follows:

d
2
3
4
5
6
7
8

#
431
787
812
399
181
195
113

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.6614821

dimension.txt.gz
The file "dimension.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 2918 records in the file.

Example record
ULID: 1FTU9VGPXXU82CTDGD6WYMBF9
Dimension: 3
Volume: 342
EhrhartDelta: [1,70,223,48]
Ehrhart: [1,74,513,...]
LogEhrhart: [0.000000000000000000000000000000,4.30406509320416975378532779249,6.24027584517076953419476314266,...]

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

For each polytope \(P\) of dimension \(d\) 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 \(2 \leq d \leq 8\) of the polytope \(P\).
Volume: A positive integer. The lattice-normalised volume \(\mathrm{Vol}(P)\) of the polytope \(P\).
EhrhartDelta: A sequence \([1,a_1,a_2,\ldots,a_d]\) of integers of length \(d + 1\). 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 \((1 + a_1t + a_2t^2 + \ldots + a_dt^d) / (1 - t)^{d+1}\). In particular, \(\mathrm{Vol}(P) = 1 + a_1 + a_2 + \ldots + a_d\).
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 \(\mathrm{Ehr}(P) = 1 + c_1t + c_2t^2 + \ldots = (1 + a_1t + a_2t^2 + \ldots + a_dt^d) / (1 - t)^{d+1}\).
LogEhrhart: A sequence \([0,y_1,y_2,\ldots,y_{1100}]\) of non-negative floating point numbers. Here \(y_i := \log c_i\)