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<p>&nbsp;<p>
<hr noshade size="5">
<h2><a name="AlffClassNumberApprox"></a>
AlffClassNumberApprox
</h2>

Compute an approximation of the class number of a global
function field.

<p><p>

<h3>
Syntax:
</h3>
<pre>hbar := AlffClassNumberApprox(F, b);</pre>

<p>

<table>
<tr>
<td>real</td>
<td><pre>  hbar  </pre></td>
<td>approximated class number</td>
</tr>
<tr>
<td>alff</td>
<td><pre>  F  </pre></td>
<td>global function field</td>
</tr>
<tr>
<td>integer</td>
<td><pre>  b  </pre></td>
<td>bound</td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#AlffClassNumberApproxBound">AlffClassNumberApproxBound</a>, <a href="kashref.html#AlffClassGroup">AlffClassGroup</a>, <a href="kashref.html#AlffLPoly">AlffLPoly</a>, <a href="kashref.html#AlffInit">AlffInit</a>, <a href="kashref.html#AlffOrders">AlffOrders</a>

<p>

<h3>
Description:
</h3>
Let F/k be a global function field of genus g over the exact
constant field of q elements. If h is the class number of F/k
we have (see He2)
\biggl| \log \bigl( h / q^g  \bigr) -
\sum_{r=1}^{b} frac{q^{-r}}{r} \bigl( N_r - (q^r + 1) \bigl)
\biggr|  &lt;=  frac{2g}{q^{1/2}-1} cdot
frac{q^{-b/2}}{b+1},  where N_r is the number of places
of degree one in the constant field extension of degree
r of F/k. By this formula an approximation of h is
computed using the values of N_r for 1  &lt;=  r  &lt;=  b.
The bound b can be determined from a given multiplicative
error bound a \in R^{>1}
via <tt> AlffClassNumberApproxBound()</tt>.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>

<pre>

</pre>

<p>


<hr>

<p>
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