<?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>poincareN -- assemble degrees into polynomial</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="___Polynomial__Ring.html">next</a> | <a href="_poincare_lp__Ring_rp.html">previous</a> | <a href="___Polynomial__Ring.html">forward</a> | <a href="_poincare_lp__Ring_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>poincareN -- assemble degrees into 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>poincareN C</tt></div>
</dd></dl>
</div>
</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>This function encodes information about the degrees of basis elements of a free chain complex in a polynomial. The polynomial has a term <tt>S^i T_0^(d_0) ... T_(n-1)^(d_(n-1))</tt> in it for each basis element of <tt>C_i</tt> with multi-degree<tt>{d_0,...,d_(n-1)}.</tt><p/>
<table class="examples"><tr><td><pre>i1 : R = ZZ/101[a,b,c, Degrees=>{1,1,2}];</pre>
</td></tr>
<tr><td><pre>i2 : C = res cokernel vars R
1 3 3 1
o2 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o2 : ChainComplex</pre>
</td></tr>
<tr><td><pre>i3 : betti C
0 1 2 3
o3 = total: 1 3 3 1
0: 1 2 1 .
1: . 1 2 1
o3 : BettiTally</pre>
</td></tr>
<tr><td><pre>i4 : p = poincareN C
2 2 2 2 3 3 4
o4 = 1 + 2S*T + S*T + S T + 2S T + S T
0 0 0 0 0
o4 : ZZ[S, T ]
0</pre>
</td></tr>
</table>
Setting the <tt>S</tt> variable to -1 gives the Poincare polynomial calculated by <a href="_poincare.html" title="assemble degrees into polynomial">poincare</a>.<table class="examples"><tr><td><pre>i5 : use ring p
o5 = ZZ[S, T ]
0
o5 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i6 : substitute(p, {S=>-1})
3 4
o6 = 1 - 2T + 2T - T
0 0 0
o6 : ZZ[S, T ]
0</pre>
</td></tr>
</table>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_poincare.html" title="assemble degrees into polynomial">poincare</a> -- assemble degrees into polynomial</span></li>
<li><span><a href="_degrees__Ring.html" title="the ring of degrees">degreesRing</a> -- the ring of degrees</span></li>
<li><span><a href="_hilbert__Function.html" title="the Hilbert function">hilbertFunction</a> -- the Hilbert function</span></li>
<li><span><a href="_hilbert__Series.html" title="compute the Hilbert series">hilbertSeries</a> -- compute the Hilbert series</span></li>
<li><span><a href="_hilbert__Polynomial.html" title="compute the Hilbert polynomial">hilbertPolynomial</a> -- compute the Hilbert polynomial</span></li>
<li><span><a href="_reduce__Hilbert.html" title="reduce a Hilbert series expression">reduceHilbert</a> -- reduce a Hilbert series expression</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>poincareN</tt> :</h2>
<ul><li>poincareN(ChainComplex)</li>
</ul>
</div>
</div>
</body>
</html>
