<?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>bFunction(Ideal,List) -- b-function 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="_b__Function_lp__Module_cm__List_cm__List_rp.html">next</a> | <a href="_b__Function_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">previous</a> | <a href="_b__Function_lp__Module_cm__List_cm__List_rp.html">forward</a> | <a href="_b__Function_lp..._cm_sp__Strategy_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>bFunction(Ideal,List) -- b-function 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>b = bFunction(I,w)</tt></div>
</dd></dl>
</div>
</li>
<li><span>Function: <a href="_b__Function.html" title="b-function">bFunction</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, a holonomic ideal in the Weyl algebra <em>A<sub>n</sub>(K)</em>.</span></li>
<li><span><tt>w</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a list of integer weights corresponding to the differential variables in the Weyl algebra.</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><tt>b</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, a polynomial <em>b(s)</em> which is the b-function of <em>I</em> with respect to <em>w</em></span></li>
</ul>
</div>
</li>
<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_b__Function_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">Strategy => ...</a>, -- specify strategy for computing b-function</span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>Use <a href="_set__Hom__Switch_lp__Boolean_rp.html" title="toggles the use of homogeneous Weyl algebra">setHomSwitch</a>(true) to force all the subroutines to use homogenized <a href="../../Macaulay2Doc/html/___Weyl__Algebra.html" title="name for an optional argument">WeylAlgebra</a><p><b>Definition. </b>The b-function <em>b(s)</em> is defined as the monic generator of the intersection of <em>in<sub>(-w,w)</sub>(I)</em> and <em>K[s]</em>, where <em>s = [w<sub>1</sub>t<sub>1</sub> + ... + w<sub>n</sub>t<sub>n</sub>]</em> (here <em>t<sub>i</sub> = x<sub>i</sub>D<sub>i</sub></em>).</p>
<table class="examples"><tr><td><pre>i1 : R = QQ[x_1,x_2,D_1,D_2,WeylAlgebra=>{x_1=>D_1,x_2=>D_2}]
o1 = R
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : I = ideal(x_1, D_2-1)
o2 = ideal (x , D - 1)
1 2
o2 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i3 : bFunction(I,{1, 0})
o3 = s + 1
o3 : QQ[s]</pre>
</td></tr>
</table>
</div>
</div>
<div class="single"><h2>Caveat</h2>
<div>The ring of I should not have any parameters: it should be a pure Weyl algebra. Similarly, this ring should not be a homogeneous <a href="../../Macaulay2Doc/html/___Weyl__Algebra.html" title="name for an optional argument">WeylAlgebra</a></div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_global__B__Function_lp__Ring__Element_rp.html" title="global b-function (else known as the Bernstein-Sato polynomial)">globalBFunction</a> -- global b-function (else known as the Bernstein-Sato polynomial)</span></li>
<li><span><a href="_factor__B__Function_lp__Ring__Element_rp.html" title="factorization of a b-function">factorBFunction</a> -- factorization of a b-function</span></li>
</ul>
</div>
</div>
</body>
</html>
