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<head><title>RatSols -- rational solutions of a holonomic system</title>
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<div><h1>RatSols -- rational 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>RatSols I, RatSols(I,f), RatSols(I,f,w), RatSols(I,ff), RatSols(I,ff,w)</tt></div>
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<li><div class="single">Inputs:<ul><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>f</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, a polynomial</span></li>
<li><span><tt>ff</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a list of polynomials</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 rational solutions of <em>I</em> with poles along <em>f</em> or along the polynomials in <tt>ff</tt> using <em>w</em> for Groebner deformations</span></li>
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<div class="single"><h2>Description</h2>
<div>The rational solutions of a holonomic system form a finite-dimensional vector space. The only possibilities for the poles of a rational solution are the codimension one components of the singular locus. An algorithm to compute rational solutions 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).<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((x+1)*D+5)
o2 = ideal(x*D + D + 5)
o2 : Ideal of W</pre>
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<tr><td><pre>i3 : RatSols I
-1
o3 = {-------------------------------}
5 4 3 2
x + 5x + 10x + 10x + 5x + 1
o3 : List</pre>
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<div class="single"><h2>Caveat</h2>
<div>The most efficient method to find rational solutions is to find the singular locus, then try to find its irreducible factors. With these, call RatSols(I, ff, w), where w should be generic enough so that the PolySols routine will not complain of a non-generic weight vector.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Poly__Sols.html" title="polynomial solutions of a holonomic system">PolySols</a> -- polynomial solutions of a holonomic system</span></li>
<li><span><a href="___Rat__Ext.html" title="Ext(holonomic D-module, polynomial ring localized at the sigular locus)">RatExt</a> -- Ext(holonomic D-module, polynomial ring localized at the sigular locus)</span></li>
<li><span><a href="___D__Hom.html" title="D-homomorphisms between holonomic D-modules">DHom</a> -- D-homomorphisms between holonomic D-modules</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>RatSols</tt> :</h2>
<ul><li>RatSols(Ideal)</li>
<li>RatSols(Ideal,List)</li>
<li>RatSols(Ideal,List,List)</li>
<li>RatSols(Ideal,RingElement)</li>
<li>RatSols(Ideal,RingElement,List)</li>
</ul>
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