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<head><title>tangentCone(Ideal)</title>
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<div><h1>tangentCone(Ideal)</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>tangentCone I</tt></div>
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<li><span>Function: <a href="_tangent__Cone_lp__Ideal_rp.html" title="">tangentCone</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, the ideal of the tangent cone of the subvariety defined by <tt>I</tt> at the point defined by the variables of the ring, with a minimal set of generators</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><tt>Strategy => </tt><span><span>default value Local</span>, <tt>Local</tt> or <tt>Global</tt></span></span></li>
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<div class="single"><h2>Description</h2>
<div><p>The tangent cone is the ideal that defines <tt>gr(R/I)</tt>, where <tt>R</tt> is the ring containing <tt>I</tt>, and <tt>gr</tt> is the associated graded ring formed with respect to maximal ideal generated by the variables.</p>
<p>The algorithm follows the method of Proposition 15.28 in the book <em>Commutative Algebra with a View Toward Algebraic Geometry</em> by David Eisenbud (Springer, Graduate Texts in Mathematics, volume 150).</p>
<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : tangentCone ideal "xz-y3,yz-x4,z2-x3y2"
2 4
o2 = ideal (z , y*z, x*z, y )
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : tangentCone ideal "z2-x5,zx-y3"
2 3 6
o3 = ideal (z , x*z, y z, y )
o3 : Ideal of R</pre>
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<tr><td><pre>i4 : tangentCone ideal "x3+x2z2,x2y+xz3+z5"
2 3 2 3 5 6 7 9
o4 = ideal (x y, x , x z , 2x*y*z - x*z , x*z , y*z )
o4 : Ideal of R</pre>
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<tr><td><pre>i5 : betti oo
0 1
o5 = total: 1 6
0: 1 .
1: . .
2: . 2
3: . .
4: . 1
5: . .
6: . 1
7: . 1
8: . .
9: . 1
o5 : BettiTally</pre>
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