<?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>degree(ProjectiveHilbertPolynomial)</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_degree_lp__Projective__Variety_rp.html">next</a> | <a href="_degree_lp__Module_rp.html">previous</a> | <a href="_degree_lp__Projective__Variety_rp.html">forward</a> | <a href="_degree_lp__Module_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>degree(ProjectiveHilbertPolynomial)</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>degree f</tt></div>
</dd></dl>
</div>
</li>
<li><span>Function: <a href="_degree.html" title="">degree</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Projective__Hilbert__Polynomial.html">projective Hilbert polynomial</a></span>, usually returned via <a href="_hilbert__Polynomial.html" title="compute the Hilbert polynomial">hilbertPolynomial</a></span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>an <a href="___Z__Z.html">integer</a></span>, the degree of any graded module having this hilbert polynomial</span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>This degree is obtained from the Hilbert polynomial <tt>f</tt> as follows: if <tt>f = d z^e/e! + lower terms in z</tt>, then <tt>d</tt> is returned. This is the lead coefficient of the highest<tt>P^e</tt> in the <a href="___Projective__Hilbert__Polynomial.html" title="the class of all Hilbert polynomials">ProjectiveHilbertPolynomial</a> display.<table class="examples"><tr><td><pre>i1 : R = QQ[a..d];</pre>
</td></tr>
<tr><td><pre>i2 : I = ideal(a^3, b^2, a*b*c);
o2 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i3 : F = hilbertPolynomial I
o3 = - 2*P + 4*P
0 1
o3 : ProjectiveHilbertPolynomial</pre>
</td></tr>
<tr><td><pre>i4 : degree F
o4 = 4</pre>
</td></tr>
</table>
The degree of this polynomial may be recovered using <a href="_dim.html" title="compute the Krull dimension">dim</a>:<table class="examples"><tr><td><pre>i5 : dim F
o5 = 1</pre>
</td></tr>
</table>
The dimension as a projective variety is also one less that the Krull dimension of <tt>R/I</tt><table class="examples"><tr><td><pre>i6 : (dim I - 1, degree I)
o6 = (1, 4)
o6 : Sequence</pre>
</td></tr>
</table>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_hilbert__Polynomial.html" title="compute the Hilbert polynomial">hilbertPolynomial</a> -- compute the Hilbert polynomial</span></li>
</ul>
</div>
</div>
</body>
</html>
