Ehrhart series coefficients for random lattice polytopes

<p><strong>Ehrhart series coefficients for random lattice polytopes</strong></p> <p>A dataset of Ehrhart data for 2918 randomly generated lattice polytopes, in dimensions 2 to 8.</p> <p>The polytopes used to generate this data were produced by the following algorithm:</p> <ol> <li>Fix <span class="math-tex">\(d\)</span> a positive integer in <span class="math-tex">\(\{2,\ldots,8\}\)</span>.</li> <li>Choose <span class="math-tex">\(d + k\)</span> lattice points <span class="math-tex">\(\{v_1,\ldots,v_{d+k}\}\)</span> uniformly at random in a box <span class="math-tex">\([-5,5]^d\)</span>, where <span class="math-tex">\(k\)</span> is chosen uniformly at random in <span class="math-tex">\(\{1,\ldots,5\}\)</span>.</li> <li>Set <span class="math-tex">\(P := \mathrm{conv}\{v_1,\ldots,v_{d+k}\}\)</span>. If <span class="math-tex">\(\mathrm{dim}(P)\neq d\)</span> then return to step 2.</li> </ol> <p>The final dataset has duplicate records removed. The data is distributed by dimension <span class="math-tex">\(d\)</span> as follows:</p> <table> <tbody> <tr> <th scope="row">d</th> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>6</td> <td>7</td> <td>8</td> </tr> <tr> <th scope="row">#</th> <td>431</td> <td>787</td> <td>812</td> <td>399</td> <td>181</td> <td>195</td> <td>113</td> </tr> </tbody> </table> <p>For details, see the paper:</p> <p>&nbsp;<em>Machine Learning the Dimension of a Polytope</em>, Tom Coates, Johannes Hofscheier, and Alexander M. Kasprzyk, 2022.</p> <p>If you make use of this data, please cite the above paper and the DOI for this data:</p> <p>&nbsp;doi:10.5281/zenodo.6614821</p> <p><strong>dimension.txt.gz</strong><br> The file &quot;dimension.txt.gz&quot; 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.</p> <p><strong>Example record</strong><br> ULID: 1FTU9VGPXXU82CTDGD6WYMBF9<br> Dimension: 3<br> Volume: 342<br> EhrhartDelta: [1,70,223,48]<br> Ehrhart: [1,74,513,...]<br> LogEhrhart: [0.000000000000000000000000000000,4.30406509320416975378532779249,6.24027584517076953419476314266,...]</p> <p>(The values for Ehrhart and LogEhrhart in the example have been truncated.)</p> <p>For each polytope <span class="math-tex">\(P\)</span> of dimension <span class="math-tex">\(d\)</span> we record the following keys and values in the dataset:</p> <p>ULID: A randomly generated string identifier for this record.<br> Dimension: A positive integer. The dimension <span class="math-tex">\(2 \leq d \leq 8\)</span> of the polytope <span class="math-tex">\(P\)</span>.<br> Volume: A positive integer. The lattice-normalised volume <span class="math-tex">\(\mathrm{Vol}(P)\)</span> of the polytope <span class="math-tex">\(P\)</span>.<br> EhrhartDelta: A sequence <span class="math-tex">\([1,a_1,a_2,\ldots,a_d]\)</span> of integers of length <span class="math-tex">\(d + 1\)</span>. This is the Ehrhart <span class="math-tex">\(\delta\)</span>-vector (or <span class="math-tex">\(h^*\)</span>-vector) of <span class="math-tex">\(P\)</span>. The Ehrhart series <span class="math-tex">\(\mathrm{Ehr}(P)\)</span> of <span class="math-tex">\(P\)</span> is given by the power-series expansion of <span class="math-tex">\((1 + a_1t + a_2t^2 + \ldots + a_dt^d) / (1 - t)^{d+1}\)</span>. In particular, <span class="math-tex">\(\mathrm{Vol}(P) = 1 + a_1 + a_2 + \ldots + a_d\)</span>.<br> Ehrhart: A sequence <span class="math-tex">\([1,c_1,c_2,\ldots,c_{1100}]\)</span> of positive integers. The value <span class="math-tex">\(c_i\)</span> is equal to the number of lattice points in the <span class="math-tex">\(i\)</span>-th dilation of <span class="math-tex">\(P\)</span>, that is, <span class="math-tex">\(c_i = \#(iP \cap \mathbb{Z}^d)\)</span>. Equivalently, <span class="math-tex">\(c_i\)</span> is the coefficient of <span class="math-tex">\(t^i\)</span> in <span class="math-tex">\(\mathrm{Ehr}(P) = 1 + c_1t + c_2t^2 + \ldots = (1 + a_1t + a_2t^2 + \ldots + a_dt^d) / (1 - t)^{d+1}\)</span>.<br> LogEhrhart: A sequence <span class="math-tex">\([0,y_1,y_2,\ldots,y_{1100}]\)</span> of non-negative floating point numbers. Here&nbsp;<span class="math-tex">\(y_i := \log c_i\)</span></p>