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<head><title>hilbertSeries(..., Order => ...) -- display the truncated power series expansion</title>
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<div><h1>hilbertSeries(..., Order => ...) -- display the truncated power series expansion</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>hilbertSeries(..., Order => n)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>n</tt>, <span>an <a href="___Z__Z.html">integer</a></span></span></li>
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<li><div class="single">Consequences:<ul><li>The output is no longer of type <a href="___Divide.html" title="the class of all divide expressions">Divide</a>. It is a polynomial in the <a href="_degrees__Ring.html">degrees ring</a>.</li>
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<div class="single"><h2>Description</h2>
<div>We compute the Hilbert series both without and with the optional argument. In the second case notice the terms of power series expansion up to, but not including, degree 5 are displayed rather than expressing the series as a rational function. The polynomial expression is an element of a Laurent polynomial ring that is the <a href="_degrees__Ring.html">degrees ring</a> of the ambient ring.<table class="examples"><tr><td><pre>i1 : R = ZZ/101[x,y];</pre>
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<tr><td><pre>i2 : hilbertSeries(R/x^3)
3
1 - T
o2 = --------
2
(1 - T)
o2 : Expression of class Divide</pre>
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<tr><td><pre>i3 : hilbertSeries(R/x^3, Order =>5)
2 3 4
o3 = 1 + 2T + 3T + 3T + 3T
o3 : ZZ[T]</pre>
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If the ambient ring is multigraded, then the <a href="_degrees__Ring.html">degrees ring</a> has multiple variables.<table class="examples"><tr><td><pre>i4 : R = ZZ/101[x,y, Degrees=>{{1,2},{2,3}}];</pre>
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<tr><td><pre>i5 : hilbertSeries(R/x^3, Order =>5)
2 2 4 2 3 3 5 4 7 4 6
o5 = 1 + T T + T T + T T + T T + T T + T T
0 1 0 1 0 1 0 1 0 1 0 1
o5 : ZZ[T , T ]
0 1</pre>
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The heft vector provides a suitable monomial ordering and degrees in the ring of the Hilbert series.<table class="examples"><tr><td><pre>i6 : R = QQ[a..d,Degrees=>{{-2,-1},{-1,0},{0,1},{1,2}}]
o6 = R
o6 : PolynomialRing</pre>
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<tr><td><pre>i7 : hilbertSeries(R, Order =>3)
2 -1 -2 -1 2 4 3 2 -1 -2
o7 = 1 + T T + T + T + T T + T T + T T + 2T + 2T T + 2T +
0 1 1 0 0 1 0 1 0 1 1 0 1 0
------------------------------------------------------------------------
-3 -1 -4 -2
T T + T T
0 1 0 1
o7 : ZZ[T , T ]
0 1</pre>
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<tr><td><pre>i8 : degrees ring oo
o8 = {{-1}, {1}}
o8 : List</pre>
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<tr><td><pre>i9 : heft R
o9 = {-1, 1}
o9 : List</pre>
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<h2>Further information</h2>
<ul><li><span>Default value: <a href="_infinity.html" title="infinity">infinity</a></span></li>
<li><span>Function: <span><a href="_hilbert__Series.html" title="compute the Hilbert series">hilbertSeries</a> -- compute the Hilbert series</span></span></li>
<li><span>Option name: <span><a href="___Order.html" title="specify the order of a Hilbert series required">Order</a> -- specify the order of a Hilbert series required</span></span></li>
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