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___Weyl__Closure.html
<!DOCTYPE html> <html lang="en"> <head> <title>WeylClosure -- Weyl closure 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="Weyl closure of an ideal" href="___Weyl__Closure.html">WeylClosure</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> next | <a href="___Walther.html">previous</a> | forward | <a href="___Walther.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>WeylClosure -- Weyl closure of an ideal</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">WeylClosure I</code></dd> <dd><code class="language-macaulay2">WeylClosure(I,f)</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 left ideal of the Weyl Algebra</span></li> <li><span><span class="tt">f</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</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>an <a title="the class of all ideals" href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, the Weyl closure (w.r.t. $f$) of $I$</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>Let $D$ be the Weyl algebra with generators $x_1,\dots,x_n$ and $\partial_1,\dots,\partial_n$ over a field $K$ of characteristic zero, and denote $R = K(x_1..x_n)<\partial_1..\partial_n>$, the ring of differential operators with rational function coefficients. The <em>Weyl closure</em> of an ideal $I$ in $D$ is the intersection of the extended ideal $R I$ with $D$. It consists of all operators which vanish on the common holomorphic solutions of $D$ and is thus analogous to the radical operation on a commutative ideal.</p> <p>The <em>partial Weyl closure</em> of $I$ with respect to a polynomial $f$ is the intersection of the extended ideal $D[f^{-1}] I$ with $D$.</p> <p>The Weyl closure is computed by localizing $D/I$ with respect to a polynomial $f$ vanishing on the singular locus, and computing the kernel of the map $D \to D/I \to (D/I)[f^{-1}]$.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : makeWA(QQ[x]) o1 = QQ[x, dx] o1 : PolynomialRing, 1 differential variable(s)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : I = ideal(x*dx-2) o2 = ideal(x*dx - 2) o2 : Ideal of QQ[x, dx]</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : holonomicRank I o3 = 1</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : WeylClosure I 3 2 o4 = ideal (x*dx - 2, x*dx - 2, dx , x*dx - dx) o4 : Ideal of QQ[x, dx]</code></pre> </td> </tr> </table> </div> <div> <h2>Caveat</h2> <div> <p>The ideal I should be of finite holonomic rank, which can be tested manually by using the function holonomicRank. The Weyl closure of non-finite rank ideals or arbitrary submodules has not been implemented.</p> </div> </div> <div> <h2>See also</h2> <ul> <li><span><a title="localization of a D-module" href="___Dlocalize.html">Dlocalize</a> -- localization of a D-module</span></li> <li><span><a title="singular locus of a D-module" href="../../WeylAlgebras/html/___Dsingular__Locus.html">DsingularLocus</a> -- singular locus of a D-module</span></li> <li><span><a title="holonomic rank of a D-module" href="../../WeylAlgebras/html/_holonomic__Rank.html">holonomicRank</a> -- holonomic rank of a D-module</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">WeylClosure</span>:</h2> <ul> <li><kbd>WeylClosure(Ideal)</kbd></li> <li><kbd>WeylClosure(Ideal,RingElement)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Weyl closure of an ideal" href="___Weyl__Closure.html">WeylClosure</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">BernsteinSato/DOC/WeylClosure.m2:53:0</span>.</p> </div> </div> </div> </body> </html>