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<head><title>stafford(Ideal) -- computes 2 generators for a given ideal in the Weyl algebra</title>
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<div><h1>stafford(Ideal) -- computes 2 generators for a given ideal in the Weyl algebra</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>stafford I</tt></div>
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<li><span>Function: <a href="_stafford_lp__Ideal_rp.html" title="computes 2 generators for a given ideal in the Weyl algebra">stafford</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, in the Weyl algebra</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>, with 2 generators (that has the same extension as I in k(x)<dx>)</span></li>
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<div class="single"><h2>Description</h2>
<div><p>A theorem of Stafford says that every ideal in the Weyl algebra can be generated by 2 elements. This routine is the implementation of the effective version of this theorem following the constructive proof in <em>A.Leykin, `Algorithmic proofs of two theorems of Stafford', Journal of Symbolic Computation, 38(6):1535-1550, 2004.</em></p>
<p>The current implementation provides a weaker result: the 2 generators produced are guaranteed to generate only the extension of the ideal <em>I</em> in the Weyl algebra with rational-function coefficients.</p>
<table class="examples"><tr><td><pre>i1 : R = QQ[x_1..x_4,D_1..D_4, WeylAlgebra=>(apply(4,i->x_(i+1)=>D_(i+1)))]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : stafford ideal (D_1,D_2,D_3,D_4)
4 2 3
o2 = ideal (D , x x D + x x D + x D + x D + D )
1 1 4 4 1 3 3 1 4 1 3 2
o2 : Ideal of QQ[x , x , x , x , D , D , D , D ]
1 2 3 4 1 2 3 4</pre>
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<div class="single"><h2>Caveat</h2>
<div>The input should be generated by at least 2 generators. The output and input ideals are not equal necessarily.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_make__Cyclic_lp__Matrix_rp.html" title="finds a cyclic generator of a D-module">makeCyclic</a> -- finds a cyclic generator of a D-module</span></li>
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