Published June 5, 2022 | Version v1
Dataset Open

Ehrhart series coefficients and quasi-period for random rational polytopes

  • 1. Imperial College London
  • 2. University of Nottingham


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:


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
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 \(y_i := \log c_i\).



Files (1.9 GB)

Name Size Download all
258.5 kB Preview Download
3.3 kB Download
1.9 GB Download
2.7 kB Download
2.0 kB Download
2.0 kB Download
2.0 kB Download
2.7 kB Download
2.0 kB Download
2.0 kB Download
2.0 kB Download
100.1 kB Preview Download
3.0 kB Preview Download

Additional details


GWT – Gromov-Witten Theory: Mirror Symmetry, Birational Geometry, and the Classification of Fano Manifolds 682603
European Commission
Classification, Computation, and Construction: New Methods in Geometry EP/N03189X/1
UK Research and Innovation