<?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(Ideal) -- assemble degrees of the quotient of the ambient ring by an ideal 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__Module_rp.html">next</a> | <a href="_poincare_lp__Chain__Complex_rp.html">previous</a> | <a href="_poincare_lp__Module_rp.html">forward</a> | <a href="_poincare_lp__Chain__Complex_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>poincare(Ideal) -- assemble degrees of the quotient of the ambient ring by an ideal 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 I</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>I</tt>, <span>an <a href="___Ideal.html">ideal</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 the quotient of the ambient ring by an ideal.<table class="examples"><tr><td><pre>i1 : R = ZZ/101[w..z];</pre> </td></tr> <tr><td><pre>i2 : I = monomialCurveIdeal(R,{1,3,4}); o2 : Ideal of R</pre> </td></tr> <tr><td><pre>i3 : poincare I 2 3 4 5 o3 = 1 - T - 3T + 4T - T o3 : ZZ[T]</pre> </td></tr> <tr><td><pre>i4 : numerator reduceHilbert hilbertSeries I 2 3 o4 = 1 + 2T + 2T - T o4 : ZZ[T]</pre> </td></tr> </table> Recall that the variables of the polynomial are the variables of the degrees ring.<table class="examples"><tr><td><pre>i5 : R=ZZ/101[x, Degrees => {{1,1}}];</pre> </td></tr> <tr><td><pre>i6 : I = ideal x^2; o6 : Ideal of R</pre> </td></tr> <tr><td><pre>i7 : poincare I 2 2 o7 = 1 - T T 0 1 o7 : ZZ[T , T ] 0 1</pre> </td></tr> <tr><td><pre>i8 : numerator reduceHilbert hilbertSeries I o8 = 1 + T T 0 1 o8 : ZZ[T , T ] 0 1</pre> </td></tr> </table> </div> </div> <div class="single"><h2>Caveat</h2> <div>As is often the case, calling this function on an ideal <tt>I</tt> actually computes it for <tt>R/I</tt> where <tt>R</tt> is the ring of <tt>I</tt>.</div> </div> </div> </body> </html>