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<head><title>reduceHilbert -- reduce a Hilbert series expression</title>
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<div><h1>reduceHilbert -- reduce a Hilbert series expression</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>reduceHilbert H</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>H</tt>, <span>a <a href="___Divide.html">divide expression</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Divide.html">divide expression</a></span>, the Hilbert series reduced by removing common factors</span></li>
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<div class="single"><h2>Description</h2>
<div><p>This function is used to reduce the rational expression given by the command <a href="_hilbert__Series.html" title="compute the Hilbert series">hilbertSeries</a>. It is not automatically reduced, but sometimes it is useful to write it in reduced form. For instance, one might not notice that the series is a polynomial until it is reduced.</p>
<table class="examples"><tr><td><pre>i1 : R = ZZ/101[x, Degrees => {2}];</pre>
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<tr><td><pre>i2 : I = ideal x^2;
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : s = hilbertSeries I
4
1 - T
o3 = --------
2
(1 - T )
o3 : Expression of class Divide</pre>
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<tr><td><pre>i4 : reduceHilbert s
2
1 + T
o4 = ------
1
o4 : Expression of class Divide</pre>
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<p>The reduction is partial, in the sense that the explicit factors of the denominator are cancelled entirely or not at all.</p>
<table class="examples"><tr><td><pre>i5 : M = R^{0,-1}
2
o5 = R
o5 : R-module, free, degrees {0, 1}</pre>
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<tr><td><pre>i6 : hilbertSeries M
1 + T
o6 = --------
2
(1 - T )
o6 : Expression of class Divide</pre>
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<tr><td><pre>i7 : f = reduceHilbert oo
1 + T
o7 = --------
2
(1 - T )
o7 : Expression of class Divide</pre>
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<tr><td><pre>i8 : gcd( value numerator f, value denominator f )
o8 = 1 + T
o8 : ZZ[T]</pre>
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<div class="waystouse"><h2>Ways to use <tt>reduceHilbert</tt> :</h2>
<ul><li>reduceHilbert(Divide)</li>
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