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<head><title>diffOps -- differential operators of up to the given order for a quotient polynomial ring</title>
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<div><h1>diffOps -- differential operators of up to the given order for a quotient polynomial ring</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>diffOps (I, k), diffOps (f, k)</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>, contained in a polynomial ring <em>R</em></span></li>
<li><span><tt>f</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, an element of a polynomial ring <em>R</em></span></li>
<li><span><tt>k</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, which is nonnegative</span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Hash__Table.html">hash table</a></span>, the differential operators of order at most <em>k</em>of the quotient ring <em>R/I</em> (or <em>R/(f)</em>)</span></li>
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<div class="single"><h2>Description</h2>
<div>Given an ideal <em>I</em> of a polynomial ring <em>R</em> the set of differential operators of the quotient ring <em>R/I</em> having order less than or equal to <em>k</em> forms a finitely generated module over <em>R/I</em>. This routine returns its generating set.<p/>
The output is in the form of a hash table. The key <tt>BasisElts</tt> is a row vector of basic differential operators. The key <tt>PolyGens</tt> is a matrix over <em>R</em> whose column vectors represent differential operators of <em>R/I</em> in the following way. For each column vector, consider its image in <tt>R/I</tt>, then take its dot product with the <tt>BasisElts</tt>. This gives a differential operator, and the set of these operators generates the differential operators of <em>R/I</em> of order <em>k</em> or less as an <em>(R/I)</em>-module.<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z]
o1 = R
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : I = ideal(x^2-y*z)
2
o2 = ideal(x - y*z)
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : diffOps(I, 3)
o3 = HashTable{BasisElts => | dx^3 dx^2dy dx^2dz dxdy^2 dxdydz dxdz^2 dy^3 dy^2dz dydz^2 dz^3 dx^2 dxdy dxdz dy^2 dydz dz^2 dx dy dz |}
PolyGens => | 0 0 0 0 0 -2xz 0 2xy yz y2 2xz 0 z2 -yz 0 0 2x2z-2yz2 |
| 0 0 0 0 0 -6yz 0 0 2xy 0 8yz y2 6xz 0 0 0 0 |
| 0 0 0 0 0 -6z2 0 12yz 4xz 6xy 4z2 -yz 0 -6xz 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 8xy 0 12yz 4y2 0 0 0 |
| 0 0 0 0 0 -24xz 0 0 8yz 0 16xz 4xy 0 -8yz 0 0 24x2z-24yz2 |
| 0 0 0 0 0 0 0 24xz 4z2 12yz 0 -4xz 0 -8z2 0 0 0 |
| 0 0 0 0 0 8y2 0 0 0 0 0 0 8xy 0 0 0 0 |
| 0 0 0 0 0 -24yz 0 0 0 0 16yz 0 0 8xy 0 0 0 |
| 0 0 0 0 0 0 0 0 8xz 0 0 4yz 0 -16xz 0 0 0 |
| 0 0 0 0 0 0 0 16z2 0 8xz 0 -4z2 0 0 0 0 0 |
| 0 z y x 0 -3z 0 -3y x 0 z y 0 -x xy xz 0 |
| 0 4x 0 2y 0 0 0 0 -2y 0 0 0 6z 8y 0 4yz 0 |
| 0 0 4x 2z 0 0 0 0 4z 6y 0 0 0 -10z 4yz 0 0 |
| 0 4y 0 0 0 24y y2 0 0 0 -12y 0 12x 0 0 4xy 0 |
| 0 0 0 4x 0 -12z -2yz 0 0 0 8z 2y 0 0 0 0 0 |
| 0 0 4z 0 0 0 z2 12z 0 12x 0 -6z 0 0 4xz 0 0 |
| 0 0 0 1 yz 0 0 0 -1 0 0 0 0 1 -y -z -3z |
| y 2 0 0 0 6 0 0 0 0 -6 0 0 0 0 0 -6x |
| -z 0 2 0 2xz 0 2z -6 0 0 0 0 0 0 0 0 0 |
o3 : HashTable</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_put__Weyl__Algebra_lp__Hash__Table_rp.html" title="transforms output of diffOps into elements of Weyl algebra">putWeylAlgebra</a> -- transforms output of diffOps into elements of Weyl algebra</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>diffOps</tt> :</h2>
<ul><li>diffOps(Ideal,ZZ)</li>
<li>diffOps(RingElement,ZZ)</li>
</ul>
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