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<!DOCTYPE html> <html lang="en"> <head> <title>mixedVolume -- computes mixed volume of a polynomial system</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 Polynomial Homotopy Continuation" href="index.html">PHCpack</a> :: <a title="computes mixed volume of a polynomial system" href="_mixed__Volume.html">mixedVolume</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="_mixed__Volume_lp..._cminteractive_eq_gt..._rp.html">next</a> | <a href="_load__Settings__Path.html">previous</a> | <a href="_mixed__Volume_lp..._cminteractive_eq_gt..._rp.html">forward</a> | <a href="_load__Settings__Path.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>mixedVolume -- computes mixed volume of a polynomial system</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">mv = mixedVolume(S)</code></dd> <dd><code class="language-macaulay2">(mv,sv) = mixedVolume(S,StableMixedVolume => true)</code></dd> <dd><code class="language-macaulay2">(mv,q,qsols) = mixedVolume(S,StartSystem => true)</code></dd> <dd><code class="language-macaulay2">(mv,sv,q,qsols) = mixedVolume(S,StableMixedVolume => true,StartSystem => true)</code></dd> <dd><code class="language-macaulay2">(mv,q,qsols) = mixedVolume(S,StartSystem => true,numThreads=4)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">S</span>, <span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, whose entries are the polynomials of a square system</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><a title="option to switch to the interactive mode of phc -m" href="_mixed__Volume_lp..._cminteractive_eq_gt..._rp.html">interactive</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value false</span>, <span>option to switch to the interactive mode of phc -m</span></span></li> <li><span><a title="option to set the number of threads when solving a start system" href="_mixed__Volume_lp..._cmnum__Threads_eq_gt..._rp.html">numThreads</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value 0</span>, <span>option to set the number of threads when solving a start system</span></span></li> <li><span><a title="optional input for computation of the stable mixed volume" href="_mixed__Volume_lp..._cm__Stable__Mixed__Volume_eq_gt..._rp.html">StableMixedVolume</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value false</span>, <span>optional input for computation of the stable mixed volume</span></span></li> <li><span><a title="optional input for computation of mixed volume by solving a random coefficient system" href="_mixed__Volume_lp..._cm__Start__System_eq_gt..._rp.html">StartSystem</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value false</span>, <span>optional input for computation of mixed volume by solving a random coefficient system</span></span></li> <li><span><a title="option to specify whether additional output is wanted" href="_mixed__Volume_lp..._cm__Verbose_eq_gt..._rp.html">Verbose</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value false</span>, <span>option to specify whether additional output is wanted</span></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">mv</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the mixed volume of the system S</span></li> <li><span><span class="tt">sv</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the stable mixed volume of the system S</span></li> <li><span><span class="tt">q</span>, <span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, whose entries are polynomials in a random coefficient system, used as a start system for the homotopy</span></li> <li><span><span class="tt">qSols</span>, <span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, whose entries are solutions of the start system q</span></li> </ul> </li> <li> <div> Consequences: <ul> <li>Writes the system to temporary files</li> <li>Invokes the command <span class="tt">phc -m</span> (with option 4)</li> <li>Stores output of phc in temporary file</li> <li>Parses and outputs the solutions.</li> </ul> </div> </li> </ul> <div> <h2>Description</h2> <div> <p>The mixed volume of a polynomial system $S:=\{f_1,\dots,f_n\}$ is defined as follows: Let $P_1,\dots,P_n$ be the Newton polytopes of $f_1,\dots,f_n$, i.e., $P_i$ is the convex hull of the exponents of the monomials in the support of $f_i$. The mixed volume of $S$ is $$ \sum_{1\leq h\leq n} \sum_{1\leq i_1\dots\leq i_h\leq n} (-1)^{n-h}V_n(P_{i_1}+\dots+P_{i_h}), $$ where $V_n$ denotes the $n$-dimensional Euclidean volume.</p> <p>Bernstein's theorem (D. N. Bernstein,<em>The number of roots of a system of equations</em>, Functional. Anal. Appl 9 (1975), no. 3, 183-185), a generalization of the classical Bezout's theorem, shows that for a zero-dimensional system, the mixed volume provides an upper bound on the number of complex isolated roots. If the coefficients of the system are sufficiently generic, the mixed volume is a sharp bound.</p> <p></p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = CC[x,y];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : f = { x^3*y^5 + y^2 + x^2*y, x*y + x^2 - 1};</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : I=ideal f; o3 : Ideal of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : dim I -- warning: experimental computation over inexact field begun -- results not reliable (one warning given per session) o4 = 0</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : degree I o5 = 10</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : m = mixedVolume(f) -- counts the number of complex roots in the torus (without zero components) o6 = 8</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : (mv,sv) = mixedVolume(f,StableMixedVolume=>true) o7 = (8, 10) o7 : Sequence</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : (mv,q,qsols) = mixedVolume(f,StartSystem=>true);</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : q --let's take a look at the start system: 3 5 2 o9 = {(- .205377 - .978683*ii)x y + (.943327 - .331864*ii)x y, (.673112 + ------------------------------------------------------------------------ 2 .739541*ii)x - .911067 - .412259*ii} o9 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : qsols --and its solutions: o10 = {{.979319-.202321*ii, .930645-.365923*ii}, {-.979319+.202321*ii, ----------------------------------------------------------------------- .916812+.399319*ii}, {.979319-.202321*ii, .365923+.930645*ii}, ----------------------------------------------------------------------- {-.979319+.202321*ii, -.399319+.916812*ii}, {.979319-.202321*ii, ----------------------------------------------------------------------- -.930645+.365923*ii}, {-.979319+.202321*ii, -.916812-.399319*ii}, ----------------------------------------------------------------------- {.979319-.202321*ii, -.365923-.930645*ii}, {-.979319+.202321*ii, ----------------------------------------------------------------------- .399319-.916812*ii}} o10 : List</code></pre> </td> </tr> </table> <div> <p><em>Note that only those solutions with nonzero components are shown, even if StableMixedVolume is true. See the end of the temporary output file for the solutions with zero components.</em></p> <p>The method <span class="tt">mixedVolume</span> calls an Ada translation of ACM TOMS Algorithm 846: <em>MixedVol: a software package for mixed-volume computation</em> by Tangan Gao, T. Y. Li, Mengnien Wu, <a href="http://portal.acm.org/citation.cfm%3Fdoid%3D1114268.1114274">ACM TOMS</a> 31(4):555-560, 2005.</p> </div> </div> <div> <h2>See also</h2> <ul> <li><span><a title="optional input for computation of the stable mixed volume" href="___Stable__Mixed__Volume.html">StableMixedVolume</a> -- optional input for computation of the stable mixed volume</span></li> <li><span><a title="optional input to construct and solve a random coefficient system" href="___Start__System.html">StartSystem</a> -- optional input to construct and solve a random coefficient system</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">mixedVolume</span>:</h2> <ul> <li><kbd>mixedVolume(List)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="computes mixed volume of a polynomial system" href="_mixed__Volume.html">mixedVolume</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">PHCpack/PHCpackDoc.m2:609:0</span>.</p> </div> </div> </div> </body> </html>