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<!DOCTYPE html> <html lang="en"> <head> <title>mMixedVolume -- Compute the mixed volume of a collection of lattice polytopes</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="Calculate mixed multiplicities, mixed volume and sectional Milnor numbers" href="index.html">MixedMultiplicity</a> :: <a title="Compute the mixed volume of a collection of lattice polytopes" href="_m__Mixed__Volume.html">mMixedVolume</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="_multi__Rees__Ideal.html">next</a> | <a href="_mixed__Multiplicity.html">previous</a> | <a href="_multi__Rees__Ideal.html">forward</a> | <a href="_mixed__Multiplicity.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>mMixedVolume -- Compute the mixed volume of a collection of lattice polytopes</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">mMixedVolume W</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">W</span>, <span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, of homogeneous ideals $I_1,\ldots,I_n$ over a polynomial ring, or a list of lists of vertices of the polytopes</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, mixed volume</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>Let $Q_1,\ldots,Q_n$ be a collection of lattice polytopes in $\mathbb{R}^n$ and let $I_1,\ldots,I_n$ be homogeneous ideals in a polynomial ring over the field of rational numbers, corresponding to the given polytopes. These ideals can be obtained using the command <a title="Compute the homogeneous ideal corresponding to the vertices of a lattice polytope in $\\mathbb{R}^n$." href="_hom__Ideal__Polytope.html">homIdealPolytope</a>. The mixed volume is calculated by computing a mixed multiplicity of these ideals.</p> <p>The following example computes the mixed volume of two 2-cross polytopes.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : I = homIdealPolytope {(-1,0),(0,-1),(1,0),(0,1)} 2 2 2 2 o1 = ideal (X X , X X , X X , X X ) 1 2 1 2 1 3 2 3 o1 : Ideal of QQ[X ..X ] 1 3</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : mMixedVolume {I,I} o2 = 4</code></pre> </td> </tr> </table> <div> <p>One can also compute the mixed volume of a collection of lattice polytopes by directly entering the vertices of the polytopes. Mixed Volume in the above example can also be computed as follows.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i3 : C = {(-1,0),(0,-1),(1,0),(0,1)} o3 = {(-1, 0), (0, -1), (1, 0), (0, 1)} o3 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : mMixedVolume {C,C} o4 = 4</code></pre> </td> </tr> </table> <div> <p>The following example computes the mixed volume of a 2-dimensional hypercube $H$ and a 2-cross polytope $C$.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i5 : H = {(1,1),(1,-1),(-1,1),(-1,-1)};</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : C = {(-1,0),(0,1),(1,0),(0,-1)};</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : mMixedVolume {H,C} o7 = 8</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="Compute the homogeneous ideal corresponding to the vertices of a lattice polytope in $\\mathbb{R}^n$." href="_hom__Ideal__Polytope.html">homIdealPolytope</a> -- Compute the homogeneous ideal corresponding to the vertices of a lattice polytope in $\mathbb{R}^n$.</span></li> <li><span><a title="Compute a given mixed multiplicity of ideals" href="_mixed__Multiplicity.html">mixedMultiplicity</a> -- Compute a given mixed multiplicity of ideals</span></li> <li><span><a title="computes the mixed volume of a list of polytope" href="../../Polyhedra/html/_mixed__Volume.html">mixedVolume</a> -- computes the mixed volume of a list of polytope</span></li> <li><span><a title="Compute the defining ideal of multi-Rees algebra of ideals" href="_multi__Rees__Ideal.html">multiReesIdeal</a> -- Compute the defining ideal of multi-Rees algebra of ideals</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">mMixedVolume</span>:</h2> <ul> <li><kbd>mMixedVolume(List)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Compute the mixed volume of a collection of lattice polytopes" href="_m__Mixed__Volume.html">mMixedVolume</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">MixedMultiplicity.m2:752:0</span>.</p> </div> </div> </div> </body> </html>