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<head><title>bFunction(Module,List,List) -- b-function of a holonomic D-module</title>
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<div><h1>bFunction(Module,List,List) -- b-function of a holonomic D-module</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(M,w,m)</tt></div>
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<li><span>Function: <a href="_b__Function.html" title="b-function">bFunction</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="../../Macaulay2Doc/html/___Module.html">module</a></span>, a holonomic module over a 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>
<li><span><tt>m</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a list of integers, each of which is the shift for the corresponding component</span></li>
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</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>M</em> with respect to <em>w</em> and <em>m</em></span></li>
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<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>
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<div class="single"><h2>Description</h2>
<div>The algorithm represents <em>M</em> as <em>F/N</em> where <em>F</em> is free and <em>N</em> is a submodule of <em>F</em>. Then it computes b-functions <em>b<sub>i</sub>(s)</em> for <em>N \cap F<sub>i</sub></em> (i-th component of <em>F</em>) and outputs <em>lcm{ b<sub>i</sub>(s-m<sub>i</sub>) }</em><table class="examples"><tr><td><pre>i1 : R = QQ[x, dx, WeylAlgebra => {x=>dx}]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : M = cokernel matrix {{x^2, 0, 0}, {0, dx^3, 0}, {0, 0, x^3}}
o2 = cokernel | x2 0 0 |
| 0 dx^3 0 |
| 0 0 x3 |
3
o2 : R-module, quotient of R</pre>
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<tr><td><pre>i3 : factorBFunction bFunction(M, {1}, {0,0,0})
o3 = (s)(s - 2)(s - 1)(s + 1)(s + 2)(s + 3)
o3 : Expression of class Product</pre>
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<tr><td><pre>i4 : factorBFunction bFunction(M, {1}, {1,2,3})
o4 = (s)(s - 4)(s - 3)(s - 2)(s - 1)(s + 1)
o4 : Expression of class Product</pre>
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<div class="single"><h2>Caveat</h2>
<div>The Weyl algebra should not have any parameters. Similarly, it should not be a homogeneous Weyl algebra</div>
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<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>
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