<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- --> <!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html --> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head><title>holonomicRank -- rank of a D-module</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="___Homology__Modules.html">next</a> | <a href="___Ground__Field.html">previous</a> | <a href="___Homology__Modules.html">forward</a> | <a href="___Ground__Field.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div> </td> </tr> </table> <hr/> <div><h1>holonomicRank -- rank of a D-module</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>holonomicRank M, holonomicRank I</tt></div> </dd></dl> </div> </li> <li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="../../Macaulay2Doc/html/___Module.html">module</a></span>, over the Weyl algebra <em>D</em></span></li> <li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, which represents the module <em>M = D/I</em></span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the rank of <em>M</em></span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div>The rank of a D-module <em>M = D^r/N</em> provides analytic information about the system of PDE's given by <em>N</em>. In particular, a theorem of Cauchy states that the dimension of holomorphic solutions to <em>N</em> in a neighborhood of a nonsinugular point is equal to the rank.<p>The rank of a D-module is defined algebraically as follows. Let <em>D</em> denote the Weyl algebra <b>C</b><tt><</tt><em>x_1,....,x_n,d_1,...,d_n</em><tt>></tt> and let <em>R</em> denote the ring of differential operators <b>C</b><tt>(</tt><em>x_1,...,x_n</em><tt>)</tt><tt><</tt><em>d_1,...,d_n</em><tt>></tt> with rational function coefficients. Then the rank of <em>M = D^r/N</em> is equal to the dimension of <em>R^r/RN</em> as a vector space over <b>C</b>(<em>x_1,...,x_n</em>).</p> <p>See the book 'Groebner deformations of hypergeometric differential equations' by Saito-Sturmfels-Takayama (1999) for more details of the algorithm.</p> <table class="examples"><tr><td><pre>i1 : W = QQ[x,y,Dx,Dy, WeylAlgebra => {x=>Dx,y=>Dy}] o1 = W o1 : PolynomialRing</pre> </td></tr> <tr><td><pre>i2 : I = ideal (x*Dx+2*y*Dy-3, Dx^2-Dy) 2 o2 = ideal (x*Dx + 2y*Dy - 3, Dx - Dy) o2 : Ideal of W</pre> </td></tr> <tr><td><pre>i3 : holonomicRank I o3 = 2</pre> </td></tr> </table> </div> </div> <div class="single"><h2>See also</h2> <ul><li><span><a href="_char__Ideal.html" title="characteristic ideal of a D-module">charIdeal</a> -- characteristic ideal of a D-module</span></li> <li><span><a href="_sing__Locus.html" title="singular locus of a D-module">singLocus</a> -- singular locus of a D-module</span></li> <li><span><a href="___Ddim.html" title="dimension of a D-module">Ddim</a> -- dimension of a D-module</span></li> </ul> </div> <div class="waystouse"><h2>Ways to use <tt>holonomicRank</tt> :</h2> <ul><li>holonomicRank(Ideal)</li> <li>holonomicRank(Module)</li> </ul> </div> </div> </body> </html>