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<head><title>globalBFunction(RingElement) -- global b-function (else known as the Bernstein-Sato polynomial)</title>
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<div><h1>globalBFunction(RingElement) -- global b-function (else known as the Bernstein-Sato 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>b = globalBFunction f</tt></div>
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<li><span>Function: <a href="_global__B__Function_lp__Ring__Element_rp.html" title="global b-function (else known as the Bernstein-Sato polynomial)">globalBFunction</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, a polynomial</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>b</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, the b-function <em>b(s)</em> in <em>Q[s]</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="_global__B__Function_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">Strategy => ...</a>, -- specify strategy for computing global b-function</span></li>
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<div class="single"><h2>Description</h2>
<div><p><b>Definition. </b>Let <em>D = A_{2n}(K) = K<x_1,...,x_n,d_1,...,d_n></em> be a Weyl algebra. The Bernstein-Sato polynomial of a polynomial f is defined to be the monic generator of the ideal of all polynomials <em>b(s)</em> in <em>K[s]</em> such that <em> b(s) f^s = Q(s,x,d) f^{s+1}</em> where <em>Q</em> lives in <em>D[s].</em></p>
<p><b>Algorithm. </b>Let <em>I_f = D<t,dt>*<t-f, d_1+df/dx_1*dt, ..., d_n+df/dx_n*dt> </em>Let <em>B(s) = bFunction(I, {1, 0, ..., 0})</em> where 1 in the weight that corresponds to <em>dt. </em>Then the global b-function is <em>b_f = B(-s-1)</em></p>
<table class="examples"><tr><td><pre>i1 : R = QQ[x]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : f = x^10
10
o2 = x
o2 : R</pre>
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<tr><td><pre>i3 : b = globalBFunction f
10 11 9 66 8 363 7 157773 6 180411 5 341693 4 16819 3
o3 = s + --s + --s + ---s + ------s + ------s + ------s + -----s +
2 5 20 10000 20000 100000 20000
------------------------------------------------------------------------
1594197 2 66429 567
--------s + -------s + -------
12500000 6250000 1562500
o3 : QQ[s]</pre>
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<tr><td><pre>i4 : factorBFunction b
1 1 2 3 4 1 3 7 9
o4 = (s + 1)(s + -)(s + -)(s + -)(s + -)(s + -)(s + --)(s + --)(s + --)(s + --)
2 5 5 5 5 10 10 10 10
o4 : Expression of class Product</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_b__Function.html" title="b-function">bFunction</a> -- b-function</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>
<li><span><a href="_general__B_lp__List_cm__Ring__Element_rp.html" title="global general Bernstein-Sato polynomial">generalB</a> -- global general Bernstein-Sato polynomial</span></li>
<li><span><a href="_global__B_lp__Ideal_cm__Ring__Element_rp.html" title="compute global b-function and b-operator for a D-module and a polynomial">globalB</a> -- compute global b-function and b-operator for a D-module and a polynomial</span></li>
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