<?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>multiplicity -- compute the Hilbert-Samuel multiplicity of an ideal</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="_normal__Cone.html">next</a> | <a href="_minimal__Reduction_lp..._cm_sp__Tries_sp_eq_gt_sp..._rp.html">previous</a> | <a href="_normal__Cone.html">forward</a> | <a href="_minimal__Reduction_lp..._cm_sp__Tries_sp_eq_gt_sp..._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>multiplicity -- compute the Hilbert-Samuel multiplicity of an ideal</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>multiplicity I</tt><br/><tt>multiplicity(I,f)</tt></div> </dd></dl> </div> </li> <li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span></span></li> <li><span><tt>f</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, optional argument, if given it should be a non-zero divisor in the ideal <tt>I</tt></span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the normalized leading coefficient of the Hilbert-Samuel polynomial of <i>I</i></span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div><div>Given an ideal <i>I⊂R</i>, “multiplicity I” returns the degree of the normal cone of <i>I</i>. When <i>R/I</i> has finite length this is the sum of the Samuel multiplicities of <i>I</i> at the various localizations of <i>R</i>. When <i>I</i> is generated by a complete intersection, this is the length of the ring <i>R/I</i> but in general it is greater. For example,</div> <table class="examples"><tr><td><pre>i1 : R=ZZ/101[x,y] o1 = R o1 : PolynomialRing</pre> </td></tr> <tr><td><pre>i2 : I = ideal(x^3, x^2*y, y^3) 3 2 3 o2 = ideal (x , x y, y ) o2 : Ideal of R</pre> </td></tr> <tr><td><pre>i3 : multiplicity I o3 = 9</pre> </td></tr> <tr><td><pre>i4 : degree I o4 = 7</pre> </td></tr> </table> </div> </div> <div class="single"><h2>Caveat</h2> <div><div/> </div> </div> <div class="waystouse"><h2>Ways to use <tt>multiplicity</tt> :</h2> <ul><li>multiplicity(Ideal)</li> <li>multiplicity(Ideal,RingElement)</li> </ul> </div> </div> </body> </html>