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<head><title>symmetricKernel(..., Variable => ...) -- Choose name for variables in the created ring</title>
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<div><h1>symmetricKernel(..., Variable => ...) -- Choose name for variables in the created 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>symmetricKernel(...,Variable=>w)</tt><br/><tt>reesIdeal(...,Variable=>w)</tt><br/><tt>reesAlgebra(...,Variable=>w)</tt><br/><tt>normalCone(...,Variable=>w)</tt><br/><tt>associatedGradedRing(...,Variable=>w)</tt><br/><tt>specialFiberIdeal(...,Variable=>w)</tt><br/><tt>specialFiber(...,Variable=>w)</tt><br/><tt>distinguished(...,Variable=>w)</tt><br/><tt>distinguishedAndMult(...,Variable=>w)</tt></div>
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<div class="single"><h2>Description</h2>
<div><div>Each of these functions creates a new ring of the form R[w<sub>0</sub>, ..., w<sub>r</sub>] or R[w<sub>0</sub>, ..., w<sub>r</sub>]/J, where R is the ring of the input ideal or module (except for <a href="_special__Fiber.html" title="special fiber of a blowup">specialFiber</a>, which creates a ring <i>K[w<sub>0</sub>, ..., w<sub>r</sub>]</i>, where <i>K</i> is the ultimate coefficient ring of the input ideal or module.) This option allows the user to change the names of the new variables in this ring. The default variable is <tt>w</tt>.</div>
<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z]/ideal(x*y^2-z^9)
o1 = R
o1 : QuotientRing</pre>
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<tr><td><pre>i2 : J = ideal(x,y,z)
o2 = ideal (x, y, z)
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : I = reesIdeal(J, Variable => p)
8 2
o3 = ideal (z*p - y*p , z*p - x*p , y*p - x*p , x*y*p - z p , x*p -
1 2 0 2 0 1 1 2 1
------------------------------------------------------------------------
7 2 2 6 3
z p , p p - z p )
2 0 1 2
o3 : Ideal of R[p , p , p ]
0 1 2</pre>
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<div>To lift the result to an ideal in a flattened ring, use <a href="../../Macaulay2Doc/html/_flatten__Ring.html" title="write a ring as a (quotient) of a polynomial ring over ZZ or a prime field">flattenRing</a>:</div>
<table class="examples"><tr><td><pre>i4 : describe ring I
o4 = R[p , p , p , Degrees => {3:{1}}, Heft => {1, 0}, MonomialOrder =>
0 1 2 {1}
------------------------------------------------------------------------
{MonomialSize => 32}, DegreeRank => 2]
{GRevLex => {3:1} }
{Position => Up }</pre>
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<tr><td><pre>i5 : I1 = first flattenRing I
9 2 8 2
o5 = ideal (- z + x*y , p z - p y, p z - p x, p y - p x, p x*y - p z , p x -
1 2 0 2 0 1 1 2 1
------------------------------------------------------------------------
2 7 2 3 6
p z , p p - p z )
2 0 1 2
o5 : Ideal of QQ[p , p , p , x, y, z]
0 1 2</pre>
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<tr><td><pre>i6 : describe ring oo
o6 = QQ[p , p , p , x..z, Degrees => {3:{1}, 3:{0}}, Heft => {0..1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 2]
0 1 2 {1} {1} {GRevLex => {3:1} }
{Position => Up }
{GRevLex => {3:1} }</pre>
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<div>Note that the rings of I and I1 both have bigradings. Use <a href="../../Macaulay2Doc/html/_new__Ring.html" title="make a copy of a ring, with some features changed">newRing</a> to make a new ring with different degrees.</div>
<table class="examples"><tr><td><pre>i7 : S = newRing(ring I1, Degrees=>{numgens ring I1:1})
o7 = S
o7 : PolynomialRing</pre>
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<tr><td><pre>i8 : describe S
o8 = QQ[p , p , p , x..z, Degrees => {6:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1]
0 1 2 {GRevLex => {3:1} }
{Position => Up }
{GRevLex => {3:1} }</pre>
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<tr><td><pre>i9 : I2 = sub(I1,vars S)
9 2 8 2
o9 = ideal (- z + x*y , p z - p y, p z - p x, p y - p x, p x*y - p z , p x -
1 2 0 2 0 1 1 2 1
------------------------------------------------------------------------
2 7 2 3 6
p z , p p - p z )
2 0 1 2
o9 : Ideal of S</pre>
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<tr><td><pre>i10 : res I2
1 7 11 6 1
o10 = S <-- S <-- S <-- S <-- S <-- 0
0 1 2 3 4 5
o10 : ChainComplex</pre>
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<h2>Further information</h2>
<ul><li><span>Default value: <tt>w</tt> (missing documentation<!-- tag: w -->)</span></li>
<li><span>Function: <span><a href="_symmetric__Kernel.html" title="Compute the Rees ring of the image of a matrix">symmetricKernel</a> -- Compute the Rees ring of the image of a matrix</span></span></li>
<li><span>Option name: <span><a href="../../Macaulay2Doc/html/___Variable.html" title="specify a name for a variable">Variable</a> -- specify a name for a variable</span></span></li>
</ul>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="../../Macaulay2Doc/html/_flatten__Ring.html" title="write a ring as a (quotient) of a polynomial ring over ZZ or a prime field">flattenRing</a> -- write a ring as a (quotient) of a polynomial ring over ZZ or a prime field</span></li>
<li><span><a href="../../Macaulay2Doc/html/_new__Ring.html" title="make a copy of a ring, with some features changed">newRing</a> -- make a copy of a ring, with some features changed</span></li>
<li><span><a href="../../Macaulay2Doc/html/_substitute.html" title="substituting values for variables">substitute</a> -- substituting values for variables</span></li>
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