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<head><title>singLocus -- singular locus of a D-module</title>
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<div><h1>singLocus -- singular locus 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>singLocus M, singLocus I</tt></div>
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<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>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, the singular locus of <em>M</em></span></li>
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<div class="single"><h2>Description</h2>
<div>The singular locus of the system of PDE's given by <em>I</em> generalizes the notion of singular point of an ODE. Geometrically, the singular locus of a D-module <em>M</em> equals the projection of the characteristic variety of <em>M</em> minus the zero section of the cotangent bundle to the base affine space <b>C</b><sup><em>n</em></sup>.<p>For details of the algorithm for computing singular locus see the book 'Groebner deformations of hypergeometric differential equations' by Saito-Sturmfels-Takayama (1999).</p>
<table class="examples"><tr><td><pre>i1 : W = QQ[x,y,Dx,Dy, WeylAlgebra => {x=>Dx,y=>Dy}]
o1 = W
o1 : PolynomialRing</pre>
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<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>
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<tr><td><pre>i3 : singLocus I
o3 = ideal(y)
o3 : Ideal of W</pre>
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<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="_holonomic__Rank.html" title="rank of a D-module">holonomicRank</a> -- rank 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>
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<div class="waystouse"><h2>Ways to use <tt>singLocus</tt> :</h2>
<ul><li>singLocus(Ideal)</li>
<li>singLocus(Module)</li>
</ul>
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