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<head><title>irreducibleCharacteristicSeries -- irreducible characteristic series of an ideal</title>
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<div><h1>irreducibleCharacteristicSeries -- irreducible characteristic series of an 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>(ics,p) = irreducibleCharacteristicSeries I</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="___Ideal.html">ideal</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>ics</tt>, <span>a <a href="___List.html">list</a></span>, a list of matrices, representing an irreducible characteristic series for <tt>I</tt></span></li>
<li><span><tt>p</tt>, <span>a <a href="___Ring__Map.html">ring map</a></span>, an isomorphism from the ring of <tt>I</tt> to the ring of the characteristic series. The ring retains the names and degrees of the variables, but reorders the variables and uses a default monomial ordering.</span></li>
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<div class="single"><h2>Description</h2>
<div><p> As we see in the example below, an irreducible characteristic series for <i>I</i> consists of a collection of triangular sets. Here, given a polynomial <i>f</i>, write <i>lvar(f)</i> for the largest variable appearing in <i>f</i> (with respect to the lexicographic order). In the example, <i>lvar(-y w+x<sup>2</sup>) = y</i> . A triangular set consists of polynomials <i>f<sub>1</sub>,…,f<sub>r</sub></i> such that <i>lvar(f<sub>1</sub>)< …< lvar(f<sub>r</sub>)</i>. In the example, <i>lvar(-x*y<sup>2</sup>+z<sup>3</sup>) = x < w = lvar(-w*y+z<sup>2</sup>)</i> . If <i>T<sub>1</sub>,…,T<sub>s</sub></i> form an irreducible characteristic series for <i>I</i> , and if <i>J<sub>i</sub></i> is the ideal generated by the largest variables of the elements of <i>T<sub>i</sub></i> , then the algebraic set <i>V(I)</i> defined by <i>I</i> is the union of the sets <i>V(T<sub>i</sub>) \V(I<sub>i</sub>)</i>, for <i>i=1,…,s</i>. The minimal associated primes of <i>I</i> can thus be recovered from the irreducible characteristic series by saturation and by throwing away superfluous primes. This is done by <a href="_minimal__Primes.html" title="minimal associated primes of an ideal">minimalPrimes</a>, which uses this routine.</p>
<table class="examples"><tr><td><pre>i1 : R = QQ[w,x,y,z];</pre>
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<tr><td><pre>i2 : (L,p) = irreducibleCharacteristicSeries ideal(x^2-y*w,x^3-z*w^2)
o2 = ({| -zw2+x3 -yw+x2 |, | x w |}, map(R,QQ[y, z, w, x],{y, z, w, x}))
o2 : Sequence</pre>
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<tr><td><pre>i3 : apply(L, m -> p m)
o3 = {| x3-w2z x2-wy |, | x w |}
o3 : List</pre>
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<tr><td><pre>i4 : p^-1
o4 = map(QQ[y, z, w, x],R,{w, x, y, z})
o4 : RingMap QQ[y, z, w, x] <--- R</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_minimal__Primes.html" title="minimal associated primes of an ideal">minimalPrimes</a> -- minimal associated primes of an ideal</span></li>
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<div class="waystouse"><h2>Ways to use <tt>irreducibleCharacteristicSeries</tt> :</h2>
<ul><li>irreducibleCharacteristicSeries(Ideal)</li>
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