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<div class="ChapSects"><a href="chap12_mj.html#X850CDAFE801E2B2A">12 <span class="Heading"> Poincare series</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12_mj.html#X7CFDEEC07F15CF82">12.1 <span class="Heading">  </span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X84117EA684724D53">12.1-1 EfficientNormalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X7EBC620581DCB4D6">12.1-2 ExpansionOfRationalFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X828B81D9829328F8">12.1-3 PoincareSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X876B3DFB7B64688C">12.1-4 PoincareSeriesPrimePart</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X7E1A4C8781A02CD0">12.1-5 PoincareSeriesLHS</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X82CBD11D84D50CBD">12.1-6 Prank</a></span>
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<h3>12 <span class="Heading"> Poincare series</span></h3>

<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>

<h4>12.1 <span class="Heading">  </span></h4>

<p><a id="X84117EA684724D53" name="X84117EA684724D53"></a></p>

<h5>12.1-1 EfficientNormalSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EfficientNormalSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EfficientNormalSubgroups</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a prime-power group <span class="SimpleMath">\(G\)</span> and, optionally, a positive integer <span class="SimpleMath">\(k\)</span>. The default is <span class="SimpleMath">\(k=4\)</span>. The function returns a list of normal subgroups <span class="SimpleMath">\(N\)</span> in <span class="SimpleMath">\(G\)</span> such that the Poincare series for <span class="SimpleMath">\(G\)</span> equals the Poincare series for the direct product <span class="SimpleMath">\((N \times (G/N))\)</span> up to degree <span class="SimpleMath">\(k\)</span>.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap11.html">1</a></span> </p>

<p><a id="X7EBC620581DCB4D6" name="X7EBC620581DCB4D6"></a></p>

<h5>12.1-2 ExpansionOfRationalFunction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExpansionOfRationalFunction</code>( <var class="Arg">f</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a positive integer <span class="SimpleMath">\(n\)</span> and a rational function <span class="SimpleMath">\(f(x)=p(x)/q(x)\)</span> where the degree of the polynomial <span class="SimpleMath">\(p(x)\)</span> is less than that of <span class="SimpleMath">\(q(x)\)</span>. It returns a list <span class="SimpleMath">\([a_0 , a_1 , a_2 , a_3 , \ldots ,a_n]\)</span> of the first <span class="SimpleMath">\(n+1\)</span> coefficients of the infinite expansion</p>

<p><span class="SimpleMath">\(f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots \)</span> .</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap8.html">1</a></span> , <span class="URL"><a href="../tutorial/chap11.html">2</a></span> </p>

<p><a id="X828B81D9829328F8" name="X828B81D9829328F8"></a></p>

<h5>12.1-3 PoincareSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">R</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">L</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite <span class="SimpleMath">\(p\)</span>-group <span class="SimpleMath">\(G\)</span> and a positive integer <span class="SimpleMath">\(n\)</span>. It returns a quotient of polynomials <span class="SimpleMath">\(f(x)=P(x)/Q(x)\)</span> whose coefficient of <span class="SimpleMath">\(x^k\)</span> equals the rank of the vector space <span class="SimpleMath">\(H_k(G,Z_p)\)</span> for all <span class="SimpleMath">\(k\)</span> in the range <span class="SimpleMath">\(k=1\)</span> to <span class="SimpleMath">\(k=n\)</span>. (The second input variable can be omitted, in which case the function tries to choose a "reasonable" value for <span class="SimpleMath">\(n\)</span>. For <span class="SimpleMath">\(2\)</span>-groups the function PoincareSeriesLHS(G) can be used to produce an <span class="SimpleMath">\(f(x)\)</span> that is correct in all degrees.)</p>

<p>In place of the group <span class="SimpleMath">\(G\)</span> the function can also input (at least <span class="SimpleMath">\(n\)</span> terms of) a minimal mod <span class="SimpleMath">\(p\)</span> resolution <span class="SimpleMath">\(R\)</span> for <span class="SimpleMath">\(G\)</span>.</p>

<p>Alternatively, the first input variable can be a list <span class="SimpleMath">\(L\)</span> of integers. In this case the coefficient of <span class="SimpleMath">\(x^k\)</span> in <span class="SimpleMath">\(f(x)\)</span> is equal to the <span class="SimpleMath">\((k+1)\)</span>st term in the list.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../tutorial/chap8.html">2</a></span> , <span class="URL"><a href="../tutorial/chap11.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArithmetic.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutModPRings.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPoincareSeries.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPoincareSeriesII.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTorAndExt.html">9</a></span> </p>

<p><a id="X876B3DFB7B64688C" name="X876B3DFB7B64688C"></a></p>

<h5>12.1-4 PoincareSeriesPrimePart</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeriesPrimePart</code>( <var class="Arg">G</var>, <var class="Arg">p</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite group <span class="SimpleMath">\(G\)</span>, a prime <span class="SimpleMath">\(p\)</span>, and a positive integer <span class="SimpleMath">\(n\)</span>. It returns a quotient of polynomials <span class="SimpleMath">\(f(x)=P(x)/Q(x)\)</span> whose coefficient of <span class="SimpleMath">\(x^k\)</span> equals the rank of the vector space <span class="SimpleMath">\(H_k(G,Z_p)\)</span> for all <span class="SimpleMath">\(k\)</span> in the range <span class="SimpleMath">\(k=1\)</span> to <span class="SimpleMath">\(k=n\)</span>.</p>

<p>The efficiency of this function needs to be improved.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../tutorial/chap8.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">3</a></span> </p>

<p><a id="X7E1A4C8781A02CD0" name="X7E1A4C8781A02CD0"></a></p>

<h5>12.1-5 PoincareSeriesLHS</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeriesLHS</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>Inputs a finite <span class="SimpleMath">\(2\)</span>-group <span class="SimpleMath">\(G\)</span> and returns a quotient of polynomials <span class="SimpleMath">\(f(x)=P(x)/Q(x)\)</span> whose coefficient of <span class="SimpleMath">\(x^k\)</span> equals the rank of the vector space <span class="SimpleMath">\(H_k(G,Z_2)\)</span> for all <span class="SimpleMath">\(k\)</span>.</p>

<p>This function was written by <strong class="button">Paul Smith</strong>. It use the Singular system for commutative algebra.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X82CBD11D84D50CBD" name="X82CBD11D84D50CBD"></a></p>

<h5>12.1-6 Prank</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Prank</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">\(p\)</span>-group <span class="SimpleMath">\(G\)</span> and returns the rank of the largest elementary abelian subgroup.</p>

<p><strong class="button">Examples:</strong></p>


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