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<!DOCTYPE html> <html lang="en"> <head> <title>diffRatFun -- derivative of a rational function in a Weyl algebra</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="derivative of a rational function in a Weyl algebra" href="_diff__Rat__Fun.html">diffRatFun</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="___Dintegration.html">next</a> | <a href="___D__Hom_lp..._cm__Strategy_eq_gt..._rp.html">previous</a> | <a href="___Dintegration.html">forward</a> | <a href="___D__Hom_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>diffRatFun -- derivative of a rational function in a Weyl algebra</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">diffRatFun(m,f)</code></dd> <dd><code class="language-macaulay2">diffRatFun(m,g,f,a)</code></dd> </dl> </li> <li>Inputs: <ul> <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>, polynomial in a Weyl algebra $D$ in $n$ variables or rational function in the fraction field of a polynomial ring in $n$ variables</span></li> <li><span><span class="tt">g</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>, polynomial in a Weyl algebra $D$ in $n$ variables</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>, of nonnegative integers $m = \{m_1,...,m_n\}$</span></li> <li><span><span class="tt">a</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, an integer</span></li> </ul> </li> <li>Outputs: <ul> <li><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>, the result of applying the product of the $(dx_i)^{m_i}$ to $f$</span></li> <li><span><span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, the result of applying the product of the $(dx_i)^{m_i}$ to $g/f^a$, written as (numerator,denominator,power of denominator)</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>Let $D$ be a Weyl algebra in the variables $x_1,..x_n$ and partials $dx_1,..,dx_n$. Let $f$ be either a polynomial or rational function in the $x_i$ and $m = (m_1,..,m_n)$ a list of nonnegative integers. The function $f$ may be given as an element of a polynomial ring in the $x_i$ or of the fraction field of that polynomial ring or of $D$. This method applies the product of the $dx_i^{m_i}$ to $f$. In the case of the input $(m,g,f,a)$, where $f \neq 0$ and $g$ are both polynomials and $a$ is a nonnegative integer, it applies the product of the $dx_i^{m_i}$ to $g/f^a$ and returns the resulting derivative as (numerator,denominator,power of denominator), not necessarily in lowest terms.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : QQ[x,y,z] o1 = QQ[x..z] o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : m = {1,1,0} o2 = {1, 1, 0} o2 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : f = x^2*y+z^5 5 2 o3 = z + x y o3 : QQ[x..z]</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : diffRatFun(m,f) o4 = 2x o4 : QQ[x..z]</code></pre> </td> </tr> </table> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i5 : makeWA(QQ[x,y,z]) o5 = QQ[x..z, dx, dy, dz] o5 : PolynomialRing, 3 differential variable(s)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : m = {1,1,0} o6 = {1, 1, 0} o6 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : f = x^2*y+z^5 5 2 o7 = z + x y o7 : QQ[x..z, dx, dy, dz]</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : diffRatFun(m,f) o8 = 2x o8 : QQ[x..z, dx, dy, dz]</code></pre> </td> </tr> </table> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i9 : frac(QQ[x,y]) o9 = frac(QQ[x..y]) o9 : FractionField</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : m = {1,2} o10 = {1, 2} o10 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : f = x/y x o11 = - y o11 : frac(QQ[x..y])</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i12 : diffRatFun(m,f) 2 o12 = -- 3 y o12 : frac(QQ[x..y])</code></pre> </td> </tr> </table> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i13 : makeWA(QQ[x,y,z]) o13 = QQ[x..z, dx, dy, dz] o13 : PolynomialRing, 3 differential variable(s)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i14 : m = {1,2,1} o14 = {1, 2, 1} o14 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i15 : g = z o15 = z o15 : QQ[x..z, dx, dy, dz]</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i16 : f = x*y o16 = x*y o16 : QQ[x..z, dx, dy, dz]</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i17 : a = 3 o17 = 3</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i18 : diffRatFun(m,g,f,a) 3 2 o18 = (-36x y , x*y, 7) o18 : Sequence</code></pre> </td> </tr> </table> </div> <div> <h2>Caveat</h2> <div> <p>Must be over a ring of characteristic $0$.</p> </div> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">diffRatFun</span>:</h2> <ul> <li><kbd>diffRatFun(List,RingElement)</kbd></li> <li><kbd>diffRatFun(List,RingElement,RingElement,ZZ)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="derivative of a rational function in a Weyl algebra" href="_diff__Rat__Fun.html">diffRatFun</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/annFs.m2:119:0</span>.</p> </div> </div> </div> </body> </html>