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

Description

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{2,,15} uniformly at random.
  3. Choose d+k lattice points {v1,,vd+k} uniformly at random in a box [5r,5r]d, where k is chosen uniformly at random in {1,,5}.
  4. Set P:=conv{v1,,vd+k}. If dim(P) is not equal to d then return to step 3.
  5. Choose a lattice point vPZd uniformly at random and replace P with the translation Pv.
  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{2,3,4} and q{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 2d4 of the polytope P.
Quasiperiod: A positive integer. The quasi-period 2q15 of the polytope P.
Volume: A positive rational number. The lattice-normalised volume Vol(P) of the polytope P.
EhrhartDelta: A sequence [1,a1,a2,,aN] of integers of length N+1, where N:=q(d+1)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++aNtN)/(1tq)d+1.
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++aNtN)/(1tq)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