<?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>hilbertBasis -- computes the Hilbert basis of a Cone</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_hirzebruch.html">next</a> | <a href="_halfspaces.html">previous</a> | <a href="_hirzebruch.html">forward</a> | <a href="_halfspaces.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>hilbertBasis -- computes the Hilbert basis of a Cone</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> HB = hilbertBasis C</tt></div>
</dd></dl>
</div>
</li>
<li><div class="single">Inputs:<ul><li><span><tt>C</tt>, <span>a <a href="___Cone.html">convex rational cone</a></span></span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, containing the elements of the Hilbert basis</span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div><p/>
The Hilbert basis of the cone <tt>C</tt> is computed by the
Project-and-Lift-algorithm by Raymond Hemmecke (see below). It computes a Hilbert basis of
the cone modulo the lineality space, so it returns the list of one column matrices that give
the Hilbert basis of the Cone if one adds the basis of the lineality space and its negative.
For the Project-and-Lift-algorithm see: <p/>
<a href="http://www.hemmecke.de/raymond/">Raymond Hemmecke's</a> <em>On the
computation of Hilbert bases of cones</em>, in A. M. Cohen, X.-S. Gao, and N. Takayama,
editors, Mathematical Software, ICMS 2002, pages 307317. World Scientific, 2002.<table class="examples"><tr><td><pre>i1 : C = posHull matrix {{1,2},{2,1}}
o1 = {ambient dimension => 2 }
dimension of lineality space => 0
dimension of the cone => 2
number of facets => 2
number of rays => 2
o1 : Cone</pre>
</td></tr>
<tr><td><pre>i2 : hilbertBasis C
o2 = {| 1 |, | 2 |, | 1 |}
| 1 | | 1 | | 2 |
o2 : List</pre>
</td></tr>
</table>
</div>
</div>
<div class="waystouse"><h2>Ways to use <tt>hilbertBasis</tt> :</h2>
<ul><li>hilbertBasis(Cone)</li>
</ul>
</div>
</div>
</body>
</html>
