One Hat Cyber Team
Your IP :
216.73.216.14
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
Eksekusi
Dir :
~
/
usr
/
share
/
doc
/
Macaulay2
/
BernsteinSato
/
html
/
View File Name :
_b__Function_lp__Module_cm__List_cm__List_rp.html
<!DOCTYPE html> <html lang="en"> <head> <title>bFunction(Module,List,List) -- b-function of a holonomic D-module</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 a holonomic D-module" href="_b__Function_lp__Module_cm__List_cm__List_rp.html">bFunction(Module,List,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__Roots_lp__Ring__Element_rp.html">next</a> | <a href="_b__Function_lp__Ideal_cm__List_rp.html">previous</a> | <a href="_b__Function__Roots_lp__Ring__Element_rp.html">forward</a> | <a href="_b__Function_lp__Ideal_cm__List_rp.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>bFunction(Module,List,List) -- b-function of a holonomic D-module</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(M,w,m)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">M</span>, <span>a <a title="the class of all modules" 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><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> <li><span><span class="tt">m</span>, <span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, a list of integers, each of which is the shift for the corresponding component</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>M</em> with respect to <em>w</em> and <em>m</em></span></li> </ul> </li> </ul> <div> <h2>Description</h2> 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><code class="language-macaulay2">i1 : R = QQ[x, dx, WeylAlgebra => {x=>dx}] o1 = R o1 : PolynomialRing, 1 differential variable(s)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">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</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">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</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">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</code></pre> </td> </tr> </table> </div> <div> <h2>Caveat</h2> The Weyl algebra should not have any parameters. Similarly, it should not be a homogeneous Weyl algebra </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 a holonomic D-module" href="_b__Function_lp__Module_cm__List_cm__List_rp.html">bFunction(Module,List,List)</a> -- b-function of a holonomic D-module</span></li> </ul> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">BernsteinSato/DOC/bFunctions.m2:117:0</span>.</p> </div> </div> </div> </body> </html>