<?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>poincare(ChainComplex) -- assemble degrees of a chain complex into a polynomial</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_poincare_lp__Ideal_rp.html">next</a> | <a href="_poincare.html">previous</a> | <a href="_poincare_lp__Ideal_rp.html">forward</a> | <a href="_poincare.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>poincare(ChainComplex) -- assemble degrees of a chain complex into 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>poincare C</tt></div>
</dd></dl>
</div>
</li>
<li><span>Function: <a href="_poincare.html" title="assemble degrees into polynomial">poincare</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>C</tt>, <span>a <a href="___Chain__Complex.html">chain complex</a></span></span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Ring__Element.html">ring element</a></span>, in the Laurent polynomial ring whose variables correspond to the degrees of the ambient ring</span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>We compute the <a href="_poincare.html">Poincare polynomial</a> of a chain complex.<table class="examples"><tr><td><pre>i1 : R = ZZ/32003[a..h];</pre>
</td></tr>
<tr><td><pre>i2 : C = res ideal(a*b, c*d, e*f)
1 3 3 1
o2 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o2 : ChainComplex</pre>
</td></tr>
<tr><td><pre>i3 : C.dd
1 3
o3 = 0 : R <---------------- R : 1
| ab cd ef |
3 3
1 : R <----------------------- R : 2
{2} | -cd -ef 0 |
{2} | ab 0 -ef |
{2} | 0 ab cd |
3 1
2 : R <--------------- R : 3
{4} | ef |
{4} | -cd |
{4} | ab |
1
3 : R <----- 0 : 4
0
o3 : ChainComplexMap</pre>
</td></tr>
<tr><td><pre>i4 : poincare C
2 4 6
o4 = 1 - 3T + 3T - T
o4 : ZZ[T]</pre>
</td></tr>
</table>
</div>
</div>
</div>
</body>
</html>
