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<head><title>der -- Module of logarithmic derivations</title>
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<div><h1>der -- Module of logarithmic derivations</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>der(A) or der(A,m)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>A</tt>, <span>a <a href="___Arrangement.html">hyperplane arrangement</a></span>, an arrangement, possibly with repeated hyperplanes</span></li>
<li><span><tt>m</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, optional list of multiplicities, one for each hyperplane</span></li>
</ul>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, A matrix whose image is the module of logarithmic derivations corresponding to the (multi)arrangement <tt>A</tt>; see below.</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="_der_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">Strategy => ...</a>, </span></li>
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<div class="single"><h2>Description</h2>
<div>The module of logarithmic derivations of an arrangement defined over a ring <tt>S</tt> is, by definition, the submodule of <tt>S</tt>-derivations with the property that <tt>D(f_i)</tt> is contained in the ideal generated by <tt>f_i</tt> for each linear form <tt>f_i</tt> in the arrangement.<p/>
More generally, if the linear form <tt>f_i</tt> is given a positive integer multiplicity <tt>m_i</tt>, then the logarithmic derivations are those <tt>D</tt> with the property that <tt>D(f_i)</tt> is in <tt>ideal(f_i^(m_i))</tt> for each linear form <tt>f_i</tt>.<p/>
The <tt>j</tt>th column of the output matrix expresses the <tt>j</tt>th generator of the derivation module in terms of its value on each linear form, in order.<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z];</pre>
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<tr><td><pre>i2 : der arrangement {x,y,z,x-y,x-z,y-z}
o2 = {1} | 1 -x+y+z -xz+z2 |
{1} | 1 z -yz+z2 |
{1} | 1 y 0 |
{1} | 1 0 0 |
{1} | 1 -x+y 0 |
{1} | 1 -x+z xy-xz-yz+z2 |
6 3
o2 : Matrix R <--- R</pre>
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<p/>
This method is implemented in such a way that any derivations of degree 0 are ignored. Equivalently, the arrangement <tt>A</tt> is forced to be essential: that is, the intersection of all the hyperplanes is the origin.<table class="examples"><tr><td><pre>i3 : prune image der typeA(3)
3
o3 = (QQ[x , x , x , x ])
1 2 3 4
o3 : QQ[x , x , x , x ]-module, free, degrees {1, 2, 3}
1 2 3 4</pre>
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<tr><td><pre>i4 : prune image der typeB(4) -- A is said to be free if der(A) is a free module
4
o4 = (QQ[x , x , x , x ])
1 2 3 4
o4 : QQ[x , x , x , x ]-module, free, degrees {1, 3, 5, 7}
1 2 3 4</pre>
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not all arrangements are free:<table class="examples"><tr><td><pre>i5 : R = QQ[x,y,z];</pre>
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<tr><td><pre>i6 : A = arrangement {x,y,z,x+y+z}
o6 = {x, y, z, x + y + z}
o6 : Hyperplane Arrangement </pre>
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<tr><td><pre>i7 : betti res prune image der A
0 1
o7 = total: 4 1
1: 1 .
2: 3 1
o7 : BettiTally</pre>
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If a list of multiplicities is not provided, the occurrences of each hyperplane are counted:<table class="examples"><tr><td><pre>i8 : R = QQ[x,y]
o8 = R
o8 : PolynomialRing</pre>
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<tr><td><pre>i9 : prune image der arrangement {x,y,x-y,y-x,y,2*x} -- rank 2 => free
2
o9 = R
o9 : R-module, free, degrees {3, 3}</pre>
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<tr><td><pre>i10 : prune image der(arrangement {x,y,x-y}, {2,2,2}) -- same thing
2
o10 = R
o10 : R-module, free, degrees {3, 3}</pre>
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<div class="waystouse"><h2>Ways to use <tt>der</tt> :</h2>
<ul><li>der(Arrangement)</li>
<li>der(Arrangement,List)</li>
</ul>
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