Published June 5, 2022 | Version v1
Dataset Open

# Ehrhart series coefficients and quasi-period for random rational polytopes

## Creators

• 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 $$r\in\{2,\ldots,15\}$$ 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:

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 $$y_i := \log c_i$$.

## Files

### Files (1.9 GB)

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## Additional details

### Funding

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