<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- -->
<!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html -->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" >
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head><title>newtonPolytope -- computes the Newton polytope of a polynomial</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_normal__Cone_lp__Polyhedron_cm__Polyhedron_rp.html">next</a> | <a href="_net_lp__Polyhedron_rp.html">previous</a> | <a href="_normal__Cone_lp__Polyhedron_cm__Polyhedron_rp.html">forward</a> | <a href="_net_lp__Polyhedron_rp.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div>
</td>
</tr>
</table>
<hr/>
<div><h1>newtonPolytope -- computes the Newton polytope of a polynomial</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>P = newtonPolytope f</tt></div>
</dd></dl>
</div>
</li>
<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span></span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><tt>P</tt>, <span>a <a href="___Polyhedron.html">convex polyhedron</a></span></span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div><p/>
The <tt>newtonPolytope</tt> of <tt>f</tt> is the convex hull of its
exponent vectors in n-space, where n is the number of variables in the ring.<p/>
Consider the Vandermond determinant in 3 variables:<table class="examples"><tr><td><pre>i1 : R = QQ[a,b,c]
o1 = R
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : f = (a-b)*(a-c)*(b-c)
2 2 2 2 2 2
o2 = a b - a*b - a c + b c + a*c - b*c
o2 : R</pre>
</td></tr>
</table>
<p/>
If we compute the Newton polytope we get a hexagon in <tt>QQ</tt>^3.<table class="examples"><tr><td><pre>i3 : P = newtonPolytope f
o3 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 6
number of rays => 0
number of vertices => 6
o3 : Polyhedron</pre>
</td></tr>
</table>
</div>
</div>
<div class="waystouse"><h2>Ways to use <tt>newtonPolytope</tt> :</h2>
<ul><li>newtonPolytope(RingElement)</li>
</ul>
</div>
</div>
</body>
</html>
