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<head><title>torusInvariants -- ring of invariants</title>
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<div><h1>torusInvariants -- ring of invariants</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>torusInvariants(T,R)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, matrix (a<sub>ij</sub>) of the action</span></li>
<li><span><span>a <a href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, the ring on which the action takes place</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 list of monomials generating the ring of invariants R<sup>T</sup></span></li>
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<div class="single"><h2>Description</h2>
<div> Let T=(K<sup>*</sup>)<sup>r</sup> be the r-dimensional torus acting on the polynomial ring R=K[X<sub>1</sub>,...,X<sub>n</sub>] diagonally. Such an action can be described as follows: there are integers a<sub>ij</sub>, i=1,...,r, j=1,...,n, such that (λ<sub>1</sub>,...,λ<sub>r</sub>)∈T acts by the substitution <br/><br/>X<sub>j</sub>→λ<sub>1</sub><sup>a<sub>1j</sub></sup>*...*λ<sub>r</sub><sup>a<sub>rj</sub></sup>X<sub>j</sub>, j=1,...,n.<br/><br/>In order to compute the ring of invariants R<sup>T</sup>, one must specify the matrix (a<sub>ij</sub>).<table class="examples"><tr><td><pre>i1 : R=QQ[x,y,z,w];</pre>
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<tr><td><pre>i2 : T=matrix({{-1,-1,2,0},{1,1,-2,-1}});
2 4
o2 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i3 : torusInvariants(T,R)
2 2
o3 = ideal (x z, x*y*z, y z)
o3 : Ideal of R</pre>
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<div class="single"><h2>Caveat</h2>
<div>It is of course possible that R<sup>T</sup>=K. At present, <tt>Normaliz cannot deal with the zero cone and will issue the (wrong) error message that the cone is not pointed.</tt></div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_val__Ring.html" title="ring of valuations">valRing</a> -- ring of valuations</span></li>
<li><span><a href="_val__Ring__Ideal.html" title="valuation ideal">valRingIdeal</a> -- valuation ideal</span></li>
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<div class="waystouse"><h2>Ways to use <tt>torusInvariants</tt> :</h2>
<ul><li>torusInvariants(Matrix,Ring)</li>
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