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<head><title>invNoetherNormalization -- Noether normalization</title>
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<div><h1>invNoetherNormalization -- Noether normalization</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>N = invNoetherNormalization I</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an object of class <a href="___Involutive__Basis.html" title="the class of all involutive bases">InvolutiveBasis</a></span></span></li>
<li><span><tt>I</tt>, <span>a <a href="../../Macaulay2Doc/html/___Groebner__Basis.html">Groebner basis</a></span></span></li>
<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><tt>N</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, the first entry of which is a list defining an invertible coordinate transformation in terms of the images of the variables of the polynomial ring; the second entry is a list of variables whose residue classes modulo the ideal in the new coordinates are (maximally) algebraically independent</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>PermuteVariables => ...</tt> (missing documentation<!-- tag: invNoetherNormalization(..., PermuteVariables => ...) -->), </span></li>
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<div class="single"><h2>Description</h2>
<div><p>invNoetherNormalization constructs an automorphism of the polynomial ring in which I defines an ideal, such that the image of I under this automorphism is in Noether normal position.</p>
<p>The automorphism is defined by the first list returned: the i-th variable of the polynomial ring is mapped to the i-th entry of that list.</p>
<p>In the new coordinates, the residue class ring is an integral ring extension of the polynomial ring in the variables given in the second list returned.</p>
<p>If the option <a href="___Permute__Variables.html" title="ensure that the last dim(I) var's are algebraically independent modulo I">PermuteVariables</a> is set to true, the second list consists of the last d variables, where d is the Krull dimension of the residue class ring.</p>
<p>Reference: D. Robertz, Noether normalization guided by monomial cone decompositions, Journal of Symbolic Computation 44, 2009, pp. 1359-1373.</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*z);
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 : N = invNoetherNormalization J
o4 = {{x, - x + y, - x + z}, {z, y}}
o4 : List</pre>
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<table class="examples"><tr><td><pre>i5 : R = QQ[w,x,y,z];</pre>
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<tr><td><pre>i6 : I = ideal(y^2*z-w*x*y^2, x*y*z-w*z^2, y^2*z-w*x^2*y*z);
o6 : Ideal of R</pre>
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<tr><td><pre>i7 : J = janetBasis I;</pre>
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<tr><td><pre>i8 : N = invNoetherNormalization J
o8 = {{w, x, - x + y, - w - x + z}, {z, y}}
o8 : List</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_janet__Basis.html" title="compute Janet basis for an ideal or a submodule of a free module">janetBasis</a> -- compute Janet basis for an ideal or a submodule of a free module</span></li>
<li><span><a href="_factor__Module__Basis.html" title="enumerate standard monomials">factorModuleBasis</a> -- enumerate standard monomials</span></li>
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<div class="waystouse"><h2>Ways to use <tt>invNoetherNormalization</tt> :</h2>
<ul><li>invNoetherNormalization(GroebnerBasis)</li>
<li>invNoetherNormalization(Ideal)</li>
<li>invNoetherNormalization(InvolutiveBasis)</li>
<li>invNoetherNormalization(Matrix)</li>
<li>invNoetherNormalization(Module)</li>
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