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<head><title>makeCyclic(Matrix) -- finds a cyclic generator of a D-module</title>
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<div><h1>makeCyclic(Matrix) -- finds a cyclic generator 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>H = makeCyclic M</tt></div>
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<li><span>Function: <a href="_make__Cyclic_lp__Matrix_rp.html" title="finds a cyclic generator of a D-module">makeCyclic</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, that specifies a map such that <tt>coker M</tt> is a holonomic D-module</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>H</tt>, <span>a <a href="../../Macaulay2Doc/html/___Hash__Table.html">hash table</a></span>, where <tt>H.Generator</tt> is a cyclic generator and <tt>H.AnnG</tt> is the annihilator ideal of this generator</span></li>
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<div class="single"><h2>Description</h2>
<div>It is proven that every holonomic module is cyclic and there is an algorithm for computing a cyclic generator.<table class="examples"><tr><td><pre>i1 : W = QQ[x, dx, WeylAlgebra => {x=>dx}]
o1 = W
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : M = matrix {{dx,0,0},{0,dx,0},{0,0,dx}} -- coker M = QQ[x]^3
o2 = | dx 0 0 |
| 0 dx 0 |
| 0 0 dx |
3 3
o2 : Matrix W <--- W</pre>
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<tr><td><pre>i3 : h = makeCyclic M
3
o3 = HashTable{AnnG => ideal(dx ) }
Generator => | x2 |
| x |
| 1 |
o3 : HashTable</pre>
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<div class="single"><h2>Caveat</h2>
<div>The module <em>M</em> must be holonomic.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_is__Holonomic.html" title="determines whether a D-module (or ideal in Weyl algebra) is holonomic">isHolonomic</a> -- determines whether a D-module (or ideal in Weyl algebra) is holonomic</span></li>
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