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<div><h1>poincare -- assemble degrees into polynomial</h1>
<div class="single"><h2>Description</h2>
<div>The Poincare polynomial is the numerator of the <a href="_hilbert__Series.html">Hilbert series</a>. It encodes information about the degrees of basis elements of given object. The polynomial has terms <tt>(-1)^i T_0^(d_0) ... T_(n-1)^(d_(n-1))</tt> in it for each basis element of <tt>C_i</tt> with multi-degree<tt>{d_0,...,d_(n-1)}</tt>. When the multi-degree has a single component, the term is<tt>(-1)^i T^(d_0).</tt><p/>
This polynomial is an element of the <a href="_degrees__Ring.html">degrees ring</a>. Notice that the monomial ordering used in the degrees ring is <tt>RevLex</tt>, so the polynomials in it will be displayed with the smallest exponents first.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_poincare__N.html" title="assemble degrees into polynomial">poincareN</a> -- assemble degrees into polynomial</span></li>
<li><span><a href="_degrees__Ring.html" title="the ring of degrees">degreesRing</a> -- the ring of degrees</span></li>
<li><span><a href="_hilbert__Function.html" title="the Hilbert function">hilbertFunction</a> -- the Hilbert function</span></li>
<li><span><a href="_hilbert__Series.html" title="compute the Hilbert series">hilbertSeries</a> -- compute the Hilbert series</span></li>
<li><span><a href="_hilbert__Polynomial.html" title="compute the Hilbert polynomial">hilbertPolynomial</a> -- compute the Hilbert polynomial</span></li>
<li><span><a href="_reduce__Hilbert.html" title="reduce a Hilbert series expression">reduceHilbert</a> -- reduce a Hilbert series expression</span></li>
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<div class="waystouse"><h2>Ways to use <tt>poincare</tt> :</h2>
<ul><li><span>poincare(BettiTally), see <span><a href="___Betti__Tally.html" title="the class of all Betti tallies">BettiTally</a> -- the class of all Betti tallies</span></span></li>
<li><span><a href="_poincare_lp__Chain__Complex_rp.html" title="assemble degrees of a chain complex into a polynomial">poincare(ChainComplex)</a> -- assemble degrees of a chain complex into a polynomial</span></li>
<li><span><a href="_poincare_lp__Ideal_rp.html" title="assemble degrees of the quotient of the ambient ring by an ideal into a polynomial">poincare(Ideal)</a> -- assemble degrees of the quotient of the ambient ring by an ideal into a polynomial</span></li>
<li><span><a href="_poincare_lp__Module_rp.html" title="assemble degrees of an module into a polynomial">poincare(Module)</a> -- assemble degrees of an module into a polynomial</span></li>
<li><span><a href="_poincare_lp__Ring_rp.html" title="assemble degrees of an ring into a polynomial">poincare(Ring)</a> -- assemble degrees of an ring into a polynomial</span></li>
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