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<head><title>PolySols -- polynomial solutions of a holonomic system</title>
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<div><h1>PolySols -- polynomial solutions of a holonomic system</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>PolySols I, PolySols M, PolySols(I,w), PolySols(M,w)</tt></div>
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</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>, holonomic ideal in the Weyl algebra <em>D</em></span></li>
<li><span><tt>w</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a weight vector</span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a basis of the polynomial solutions of <em>I</em>(or of D-homomorhpisms between <em>M</em> and the polynomial ring) using <em>w</em> for Groebner deformations</span></li>
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<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="___Poly__Sols_lp..._cm_sp__Alg_sp_eq_gt_sp..._rp.html">Alg => ...</a>, -- algorithm for finding polynomial solutions</span></li>
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<div class="single"><h2>Description</h2>
<div>The polynomial solutions of a holonomic system form a finite-dimensional vector space. There are two algorithms implemented to get these solutions. The first algorithm is based on Groebner deformations and works for ideals <em>I</em> of PDE's -- see the paper 'Polynomial and rational solutions of a holonomic system' by Oaku-Takayama-Tsai (2000). The second algorithm is based on homological algebra -- see the paper 'Computing homomorphims between holonomic D-modules' by Tsai-Walther (2000).<table class="examples"><tr><td><pre>i1 : W = QQ[x, D, WeylAlgebra=>{x=>D}]
o1 = W
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : I = ideal(D^2, (x-1)*D-1)
2
o2 = ideal (D , x*D - D - 1)
o2 : Ideal of W</pre>
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<tr><td><pre>i3 : PolySols I
o3 = {x - 1}
o3 : List</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Rat__Sols.html" title="rational solutions of a holonomic system">RatSols</a> -- rational solutions of a holonomic system</span></li>
<li><span><a href="___Dintegration.html" title="integration modules of a D-module">Dintegration</a> -- integration modules of a D-module</span></li>
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<div class="waystouse"><h2>Ways to use <tt>PolySols</tt> :</h2>
<ul><li>PolySols(Ideal)</li>
<li>PolySols(Ideal,List)</li>
<li>PolySols(Module)</li>
<li>PolySols(Module,List)</li>
</ul>
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