Published June 5, 2022 | Version v1
Dataset Open

Ehrhart series coefficients for random lattice polytopes

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

Description

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:

  1. Fix d a positive integer in {2,,8}.
  2. Choose d+k lattice points {v1,,vd+k} uniformly at random in a box [5,5]d, where k is chosen uniformly at random in {1,,5}.
  3. Set P:=conv{v1,,vd+k}. If dim(P)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 2d8 of the polytope P.
Volume: A positive integer. The lattice-normalised volume Vol(P) of the polytope P.
EhrhartDelta: A sequence [1,a1,a2,,ad] of integers of length d+1. This is the Ehrhart δ-vector (or h-vector) of P. The Ehrhart series Ehr(P) of P is given by the power-series expansion of (1+a1t+a2t2++adtd)/(1t)d+1. In particular, Vol(P)=1+a1+a2++ad.
Ehrhart: A sequence [1,c1,c2,,c1100] of positive integers. The value ci is equal to the number of lattice points in the i-th dilation of P, that is, ci=#(iPZd). Equivalently, ci is the coefficient of ti in Ehr(P)=1+c1t+c2t2+=(1+a1t+a2t2++adtd)/(1t)d+1.
LogEhrhart: A sequence [0,y1,y2,,y1100] of non-negative floating point numbers. Here yi:=logci

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Additional details

Funding

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