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<head><title>charIdeal -- characteristic ideal of a D-module</title>
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<div><h1>charIdeal -- characteristic ideal 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>charIdeal M, charIdeal 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 characteristic ideal of <em>M</em></span></li>
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<div class="single"><h2>Description</h2>
<div>The characteristic ideal of <em>M</em> is the annihilator of <em>gr(M)</em> under a good filtration with respect to the order filtration. If <em>D</em> is the Weyl algebra <b>C</b><tt><</tt><em>x_1,....,x_n,d_1,...,d_n</em><tt>></tt>, then the order filtration corresponds to the weight vector (0,...,0,1...,1). The characteristic ideal lives in the associated graded ring of <em>D</em> with respect to the order filtration, and this is a commutative polynomial ring <b>C</b><tt>[</tt><em>x_1,....,x_n,xi_1,...,xi_n</em><tt>]</tt> -- here the <em>xi</em>'s are the symbols of the <em>d</em>'s. The zero locus of the characteristic ideal is equal to the characteristic variety of <em>D/I</em>, which is an invariant of a D-module.<p>The algorithm to compute the characteristic ideal consists of computing the initial ideal of I with respect to the weight vector (0,...,0,1...,1). See the book 'Groebner deformations of hypergeometric differential equations' by Saito-Sturmfels-Takayama (1999) for more details.</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 : charIdeal I
2
o3 = ideal (Dx , x*Dx + 2y*Dy)
o3 : Ideal of QQ[x, y, Dx, Dy]</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Ddim.html" title="dimension of a D-module">Ddim</a> -- dimension 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="_holonomic__Rank.html" title="rank of a D-module">holonomicRank</a> -- rank of a D-module</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>charIdeal</tt> :</h2>
<ul><li>charIdeal(Ideal)</li>
<li>charIdeal(Module)</li>
</ul>
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