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<head><title>inw -- initial form/ideal w.r.t. a weight</title>
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<div><h1>inw -- initial form/ideal w.r.t. a weight</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>inF = inw(F,w), inI = inw(I,w), inM = inw(M,w)</tt></div>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>F</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, an element of the Weyl algebra</span></li>
<li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, in the Weyl algebra</span></li>
<li><span><tt>M</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, with entries in the Weyl algebra</span></li>
<li><span><tt>w</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, of weights</span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><tt>inF</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, the initial form of <em>F</em> with respect to the weight vector</span></li>
<li><span><tt>inI</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, the initial ideal of <em>I</em> with respect to the weight vector</span></li>
<li><span><tt>inM</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, with the columns generating the initial module of the image of <em>M</em> with respect to the weight vector</span></li>
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</li>
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<div class="single"><h2>Description</h2>
<div>This routine computes the initial ideal of a left ideal <em>I</em> of the Weyl algebra with respect to a weight vector <em>w = (u,v)</em> where <em>u+v >= 0</em>. In the case where u+v > 0, then the ideal lives in the associated graded ring which is a commutative ring. In the case where u+v = 0, then the ideal lives in the associated graded ring which is again the Weyl algebra. In the general case <em>u+v >= 0</em> the associated graded ring is somewhere between. There are two strategies to compute the initial ideal. One is to homogenize to an ideal of the homogeneous Weyl algebra. The other is to homogenize with respect to the weight vector <em>w</em>.<table class="examples"><tr><td><pre>i1 : W = QQ[x,y,Dx,Dy, WeylAlgebra => {x=>Dx,y=>Dy}]
o1 = W
o1 : PolynomialRing</pre>
</td></tr>
<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>
</td></tr>
<tr><td><pre>i3 : inw(I, {1,3,3,-1})
2 2 2
o3 = ideal (x*Dx, 4y Dy + 2x*Dx, 2y*Dx*Dy, Dx )
o3 : Ideal of QQ[x, y, Dx, Dy]</pre>
</td></tr>
<tr><td><pre>i4 : inw(I, {-1,-3,1,3})
o4 = ideal (x*Dx + 2y*Dy - 3, Dy)
o4 : Ideal of W</pre>
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<div class="single"><h2>Caveat</h2>
<div>The weight vector <em>w = (u,v)</em> must have <em>u+v>=0</em>.</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_gbw.html" title="Groebner basis w.r.t. a weight">gbw</a> -- Groebner basis w.r.t. a weight</span></li>
<li><span><a href="_set__Hom__Switch_lp__Boolean_rp.html" title="toggles the use of homogeneous Weyl algebra">setHomSwitch</a> -- toggles the use of homogeneous Weyl algebra</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>inw</tt> :</h2>
<ul><li>inw(Ideal,List)</li>
<li>inw(Matrix,List)</li>
<li>inw(RingElement,List)</li>
</ul>
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