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<head><title>PermuteVariables -- ensure that the last dim(I) var's are algebraically independent modulo I</title>
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<div><h1>PermuteVariables -- ensure that the last dim(I) var's are algebraically independent modulo I</h1>
<div class="single"><h2>Description</h2>
<div><p>The symbol PermuteVariables is an option for <a href="_inv__Noether__Normalization.html" title="Noether normalization">invNoetherNormalization</a>.</p>
<p>The default value for this option is false. If set to true, the second list of the result of <a href="_inv__Noether__Normalization.html" title="Noether normalization">invNoetherNormalization</a> consists of the last d variables in the new coordinates, where d is the Krull dimension of the ring under consideration.</p>
<p>In the new coordinates defined by <a href="_inv__Noether__Normalization.html" title="Noether normalization">invNoetherNormalization</a> the residue class ring is an integral ring extension of the polynomial ring in the last d variables.</p>
<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z];</pre>
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<tr><td><pre>i2 : I = ideal(x*y^2+2*x^2*y, z^3);
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : J = janetBasis I;</pre>
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<tr><td><pre>i4 : N1 = invNoetherNormalization J
o4 = {{x, - x + y, z}, {y}}
o4 : List</pre>
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<tr><td><pre>i5 : N2 = invNoetherNormalization(J, PermuteVariables => true)
o5 = {{x, - x + z, y}, {z}}
o5 : List</pre>
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<table class="examples"><tr><td><pre>i6 : R = QQ[w,x,y,z];</pre>
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<tr><td><pre>i7 : I = ideal(w*x-y^2, y*z-x^2)
2 2
o7 = ideal (w*x - y , - x + y*z)
o7 : Ideal of R</pre>
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<tr><td><pre>i8 : J = janetBasis I;</pre>
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<tr><td><pre>i9 : N1 = invNoetherNormalization J
o9 = {{w, x, y, z}, {z, w}}
o9 : List</pre>
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<tr><td><pre>i10 : J1 = janetBasis substitute(gens I, for i in toList(0..numgens(R)-1) list R_i => N1#0#i);</pre>
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<tr><td><pre>i11 : F1 = factorModuleBasis(J1)
+----+------+
o11 = |1 |{z} |
+----+------+
|y |{z} |
+----+------+
| 2 | |
|y |{z} |
+----+------+
| 3 | |
|y |{z} |
+----+------+
|x |{z} |
+----+------+
|x*y |{z} |
+----+------+
|w |{z, w}|
+----+------+
|w*y |{z, w}|
+----+------+
| 2| |
|w*y |{z, w}|
+----+------+
| 3| |
|w*y |{z, w}|
+----+------+
o11 : FactorModuleBasis</pre>
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<tr><td><pre>i12 : N2 = invNoetherNormalization(J, PermuteVariables => true)
o12 = {{y, x, w, z}, {z, y}}
o12 : List</pre>
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<tr><td><pre>i13 : J2 = janetBasis substitute(gens I, for i in toList(0..numgens(R)-1) list R_i => N2#0#i);</pre>
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<tr><td><pre>i14 : F2 = factorModuleBasis(J2)
+---+------+
o14 = |1 |{z, y}|
+---+------+
|x |{z, y}|
+---+------+
|w |{z, y}|
+---+------+
|w*x|{z, y}|
+---+------+
o14 : FactorModuleBasis</pre>
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<div class="single"><h2>Functions with optional argument named PermuteVariables :</h2>
<ul><li><span><tt>invNoetherNormalization(..., PermuteVariables => ...)</tt> (missing documentation<!-- tag: [invNoetherNormalization, PermuteVariables] -->)</span></li>
</ul>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Permute__Variables.html" title="ensure that the last dim(I) var's are algebraically independent modulo I">PermuteVariables</a> is <span>a <a href="../../Macaulay2Doc/html/___Symbol.html">symbol</a></span>.</p>
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