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<!DOCTYPE html> <html lang="en"> <head> <title>slackIdeal -- computes the slack ideal</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="a package for slack ideals of polytopes and matroids" href="index.html">SlackIdeals</a> :: <a title="computes the slack ideal" href="_slack__Ideal.html">slackIdeal</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="_slack__Ideal_lp..._cm__Coefficient__Ring_eq_gt..._rp.html">next</a> | <a href="_slack__From__Plucker_lp..._cm__Object_eq_gt..._rp.html">previous</a> | <a href="_slack__Ideal_lp..._cm__Coefficient__Ring_eq_gt..._rp.html">forward</a> | <a href="_slack__From__Plucker_lp..._cm__Object_eq_gt..._rp.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>slackIdeal -- computes the slack ideal</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">I = slackIdeal(d, V)</code></dd> <dd><code class="language-macaulay2">I = slackIdeal M</code></dd> <dd><code class="language-macaulay2">I = slackIdeal P</code></dd> <dd><code class="language-macaulay2">I = slackIdeal C</code></dd> <dd><code class="language-macaulay2">I = slackIdeal V</code></dd> <dd><code class="language-macaulay2">I = slackIdeal(d, S)</code></dd> <dd><code class="language-macaulay2">I = slackIdeal S</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">d</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the dimension of the polytope or 1 less than the rank of the matroid</span></li> <li><span><span class="tt">V</span>, <span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, a list of vertices of a polytope, the vectors of a linear matroid, or facet sets of an abstract polytope</span></li> <li><span><span class="tt">M</span>, <span>a <a title="the class of all matroids" href="../../Matroids/html/___Matroid.html">matroid</a></span>, a matroid</span></li> <li><span><span class="tt">P</span>, <span>a <a title="the class of all convex polyhedra" href="../../Polyhedra/html/___Polyhedron.html">convex polyhedron</a></span>, a polytope</span></li> <li><span><span class="tt">C</span>, <span>a <a title="the class of all rational convex polyhedral cones" href="../../Polyhedra/html/___Cone.html">convex rational cone</a></span>, a cone</span></li> <li><span><span class="tt">S</span>, <span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, a (symbolic) slack matrix</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><a title="specifies the coefficient ring of the underlying ring of the ideal" href="_slack__Ideal_lp..._cm__Coefficient__Ring_eq_gt..._rp.html">CoefficientRing</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value QQ</span>, <span>specifies the coefficient ring of the underlying ring of the ideal</span></span></li> <li><span><a title="specify combinatorial object" href="_slack__Ideal_lp..._cm__Object_eq_gt..._rp.html">Object</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value "polytope"</span>, <span>specify combinatorial object</span></span></li> <li><span><a title="specifies saturation strategy to be used" href="_slack__Ideal_lp..._cm__Saturate_eq_gt..._rp.html">Saturate</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value "all"</span>, <span>specifies saturation strategy to be used</span></span></li> <li><span><a title="specifies saturation strategy to be used" href="_slack__Ideal_lp..._cm__Strategy_eq_gt..._rp.html">Strategy</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value Eliminate</span>, <span>specifies saturation strategy to be used</span></span></li> <li><span><a title="specifies the variables to use to create the underlying ring of the ideal" href="_slack__Ideal_lp..._cm__Vars_eq_gt..._rp.html">Vars</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value {}</span>, <span>specifies the variables to use to create the underlying ring of the ideal</span></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">I</span>, <span>an <a title="the class of all ideals" href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, the slack ideal - the saturated ideal of (d+2)-minors of the symbolic slack matrix</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>The slack ideal of a d-polytope or rank d+1 matroid is the ideal of (d+2)-minors of its symbolic slack matrix, saturated by the product of the variables in the matrix.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : V = {{0, 0}, {1, 0}, {1, 1}, {0, 1}};</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : I = slackIdeal V Order of vertices is {{0, 0}, {1, 0}, {0, 1}, {1, 1}} o2 = ideal(x x x x - x x x x ) 0 3 5 6 1 2 4 7 o2 : Ideal of QQ[x ..x ] 0 7</code></pre> </td> </tr> </table> <div> <p>If a list of points is given it can be treated as the vertices of a polytope, the ground set of a matroid or the facets of an abstract polytope by specifying the option <a title="select the combinatorial object which the input should be interpreted as" href="___Object.html">Object</a>. The default is as a polytope.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i3 : V = {{0, 0}, {1, 0}, {1, 1}, {0, 1}};</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : IP = slackIdeal V Order of vertices is {{0, 0}, {1, 0}, {0, 1}, {1, 1}} o4 = ideal(x x x x - x x x x ) 0 3 5 6 1 2 4 7 o4 : Ideal of QQ[x ..x ] 0 7</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : IM = slackIdeal(V, Object => "matroid") o5 = ideal (x x x + x x x , x x x + x x x , x x x + x x x , x x x + 4 8 10 5 7 11 1 8 9 2 6 11 0 5 9 2 3 10 0 4 6 ------------------------------------------------------------------------ x x x , x x x x - x x x x , x x x x - x x x x , x x x x - 1 3 7 1 3 8 10 0 5 6 11 0 4 8 9 2 3 7 11 1 5 7 9 ------------------------------------------------------------------------ x x x x ) 2 4 6 10 o5 : Ideal of QQ[x ..x ] 0 11</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="computes the slack matrix of a given realization" href="_slack__Matrix.html">slackMatrix</a> -- computes the slack matrix of a given realization</span></li> <li><span><a title="computes the symbolic slack matrix" href="_symbolic__Slack__Matrix.html">symbolicSlackMatrix</a> -- computes the symbolic slack matrix</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">slackIdeal</span>:</h2> <ul> <li><kbd>slackIdeal(Cone)</kbd></li> <li><kbd>slackIdeal(List)</kbd></li> <li><kbd>slackIdeal(Matrix)</kbd></li> <li><kbd>slackIdeal(Matroid)</kbd></li> <li><kbd>slackIdeal(Polyhedron)</kbd></li> <li><kbd>slackIdeal(ZZ,List)</kbd></li> <li><kbd>slackIdeal(ZZ,Matrix)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="computes the slack ideal" href="_slack__Ideal.html">slackIdeal</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function__With__Options.html">method function with options</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">SlackIdeals.m2:1924:0</span>.</p> </div> </div> </div> </body> </html>