One Hat Cyber Team

One Hat Cyber Team

Your IP : 216.73.216.80

Server IP : 194.44.31.54

Server : Linux zen.imath.kiev.ua 4.18.0-553.77.1.el8_10.x86_64 #1 SMP Fri Oct 3 14:30:23 UTC 2025 x86_64

Server Software : Apache/2.4.37 (Rocky Linux) OpenSSL/1.1.1k

PHP Version : 5.6.40

Buat File | Buat Folder

Dir : ~/usr/share/doc/Macaulay2/BernsteinSato/html/

Edit File: _b__Function_lp__Ideal_cm__List_rp.html

<!DOCTYPE html>
<html lang="en">
  <head>
    <title>bFunction(Ideal,List) -- b-function of an ideal</title>
    <meta content="text/html; charset=utf-8" http-equiv="Content-Type">
    <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css">

<link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css">
    <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script>
    <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script>
    <script>
      var macros = {
          "\\break": "\\\\",
          "\\ZZ": "\\mathbb{Z}",
          "\\NN": "\\mathbb{N}",
          "\\QQ": "\\mathbb{Q}",
          "\\RR": "\\mathbb{R}",
          "\\CC": "\\mathbb{C}",
          "\\PP": "\\mathbb{P}"
      }, delimiters = [
          { left: "$$",  right: "$$",  display: true},
          { left: "\\[", right: "\\]", display: true},
          { left: "$",   right: "$",   display: false},
          { left: "\\(", right: "\\)", display: false}
      ], ignoredTags = [
	  "kbd", "var", "samp", "script", "noscript",
	  "style", "textarea", "pre", "code", "option" ];
      document.addEventListener("DOMContentLoaded", function() {
        renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true });
      });
    </script>
    <style>.katex { font-size: 1em; }</style>
    <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script>
    <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script>
    <script src="../../../../Macaulay2/Style/prism.js"></script>
    <script>var current_version = '1.25.06';</script>
    <script src="../../../../Macaulay2/Style/version-select.js"></script>
    <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif">
  </head>
  <body>
    <div id="buttons">
      <div>
<a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="algorithms for b-functions, local cohomology, and intersection cohomology" href="index.html">BernsteinSato</a> :: <a title="b-function of an ideal" href="_b__Function_lp__Ideal_cm__List_rp.html">bFunction(Ideal,List)</a></span>      </div>
      <div class="right">
<form method="get" action="https://www.google.com/search">
  <input placeholder="Search" type="text" name="q" value="">
  <input type="hidden" name="q" value="site:macaulay2.com/doc">
</form>
<a href="_b__Function_lp__Module_cm__List_cm__List_rp.html">next</a> | <a href="_b__Function_lp..._cm__Strategy_eq_gt..._rp.html">previous</a> | <a href="_b__Function_lp__Module_cm__List_cm__List_rp.html">forward</a> | <a href="_b__Function_lp..._cm__Strategy_eq_gt..._rp.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a>      </div>
    </div>
    <hr>
    <div>
      <h1>bFunction(Ideal,List) -- b-function of an ideal</h1>
      <ul>
        <li><span>Function: <a title="b-function" href="_b__Function.html">bFunction</a></span></li>
        <li>
          <dl class="element">
            <dt>Usage: </dt>
            <dd><code class="language-macaulay2">b = bFunction(I,w)</code></dd>
          </dl>
        </li>
        <li>Inputs:          <ul>
            <li><span><span class="tt">I</span>, <span>an <a title="the class of all ideals" 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><span class="tt">w</span>, <span>a <a title="the class of all lists -- {...}" 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>
        </li>
        <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:          <ul>
            <li><span><a title="specify strategy for computing b-function" href="_b__Function_lp..._cm__Strategy_eq_gt..._rp.html">Strategy</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value IntRing</span>, <span>specify strategy for computing b-function</span></span></li>
          </ul>
        </li>
        <li>Outputs:          <ul>
            <li><span><span class="tt">b</span>, <span>a <a title="the class of all ring elements handled by the engine" 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>
        </li>
      </ul>
      <div>
        <h2>Description</h2>
Use <a title="toggle the use of homogeneous Weyl algebra" href="../../WeylAlgebras/html/_set__Hom__Switch.html">setHomSwitch</a>(true) to force all the subroutines to use homogenized <a title="specify differential operators in the ring" href="../../Macaulay2Doc/html/_monoid_lp..._cm__Weyl__Algebra_eq_gt..._rp.html">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><code class="language-macaulay2">i1 : R = QQ[x_1,x_2,D_1,D_2,WeylAlgebra=>{x_1=>D_1,x_2=>D_2}]

o1 = R

o1 : PolynomialRing, 2 differential variable(s)</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i2 : I = ideal(x_1, D_2-1)

o2 = ideal (x , D  - 1)
             1   2

o2 : Ideal of R</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i3 : bFunction(I,{1, 0})

o3 = s + 1

o3 : QQ[s]</code></pre>
            </td>
          </tr>
        </table>
      </div>
      <div>
        <h2>Caveat</h2>
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 title="specify differential operators in the ring" href="../../Macaulay2Doc/html/_monoid_lp..._cm__Weyl__Algebra_eq_gt..._rp.html">WeylAlgebra</a>      </div>
      <div>
        <h2>See also</h2>
        <ul>
          <li><span><a title="global b-function (else known as the Bernstein-Sato polynomial)" href="_global__B__Function_lp__Ring__Element_rp.html">globalBFunction</a> -- global b-function (else known as the Bernstein-Sato polynomial)</span></li>
          <li><span><a title="factorization of a b-function" href="_factor__B__Function_lp__Ring__Element_rp.html">factorBFunction</a> -- factorization of a b-function</span></li>
        </ul>
      </div>
      <div>
        <div class="waystouse">
          <h2>Ways to use this method:</h2>
          <ul>
            <li><span><a title="b-function of an ideal" href="_b__Function_lp__Ideal_cm__List_rp.html">bFunction(Ideal,List)</a> -- b-function of an ideal</span></li>
          </ul>
        </div>
        <hr>
        <div class="waystouse">
          <p>The source of this document is in <span class="tt">BernsteinSato/DOC/bFunctions.m2:80:0</span>.</p>
        </div>
      </div>
    </div>
  </body>

</html>