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

Computes the Puiseux expansions in T^{-1/e} of the roots of the
defining polynomial of a global function field F/FF_q(T).

<p><p>

<h3>
Syntax:
</h3>
<pre>L := AlffRoots(F);</pre>

<p>

<table>
<tr>
<td>list</td>
<td><pre>  L  </pre></td>
<td></td>
</tr>
<tr>
<td>global function field</td>
<td><pre>  F  </pre></td>
<td></td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#AlffRootParams">AlffRootParams</a>, <a href="kashref.html#InftyVal">InftyVal</a>

<p>

<h3>
Description:
</h3>
If the infinite place \infty of k(T) is tamely ramified in F,
then all roots of the defining
polynomial f of F/FF_q(T) can be expanded in T^{-1/e}.
The function returns a list

L:=[rho_{1,1}, &hellip; ,rho_{1,e_1 f_1},rho_{2,1}, &hellip; ,
rho_{s,e_s f_s}],

where rho_{i,j}\inFF_{q^d}\langle T^{-1/e_i}rangle\subset
FF_{q^d}\langle T^{-1/e}rangle, (1 &lt;=  j &lt;=  e_i f_i), are
the root expansions at the s infinite places above \infty
(the e_i and f_i are the ramification indices and residue
degrees of these places, e is the lcm of the e_i).

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Compute all roots of y^3+T^4+1 over FF_25:
<pre>

kash> AlffInit(FF(5,2));
"Defining global variables: k, w, kT, kTf, kTy, T, y, AlffGlobals"
kash> AlffOrders(y^3+T^4+1);
"Defining global variables: F, o, oi, one"
kash> AlffRoots(F);
> [ 4*(T^(-1/3))^-4 + 3*(T^(-1/3))^8 + 4*(T^(-1/3))^20 + 2*(T^(-1/3))^56 + 4*(T^\
(-1/3))^68 + 2*(T^(-1/3))^80 + 2*(T^(-1/3))^116 + 4*(T^(-1/3))^128 + 2*(T^(-1/\
3))^140 + 4*(T^(-1/3))^176 + 3*(T^(-1/3))^188 + 4*(T^(-1/3))^200 + 4*(T^(-1/3)\
)^296 + 3*(T^(-1/3))^308 + 4*(T^(-1/3))^320 + 2*(T^(-1/3))^356 + 4*(T^(-1/3))^\
368 + 2*(T^(-1/3))^380 + 2*(T^(-1/3))^416 + 4*(T^(-1/3))^428 + 2*(T^(-1/3))^44\
0 + 4*(T^(-1/3))^476 + 3*(T^(-1/3))^488 + 4*(T^(-1/3))^500 + 2*(T^(-1/3))^1496\
 + O((T^(-1/3))^1498), 
  w^20*(T^(-1/3))^-4 + w^2*(T^(-1/3))^8 + w^20*(T^(-1/3))^20 + w^14*(T^(-1/3))\
^56 + w^20*(T^(-1/3))^68 + w^14*(T^(-1/3))^80 + w^14*(T^(-1/3))^116 + w^20*(T^\
(-1/3))^128 + w^14*(T^(-1/3))^140 + w^20*(T^(-1/3))^176 + w^2*(T^(-1/3))^188 +\
 w^20*(T^(-1/3))^200 + w^20*(T^(-1/3))^296 + w^2*(T^(-1/3))^308 + w^20*(T^(-1/\
3))^320 + w^14*(T^(-1/3))^356 + w^20*(T^(-1/3))^368 + w^14*(T^(-1/3))^380 + w^\
14*(T^(-1/3))^416 + w^20*(T^(-1/3))^428 + w^14*(T^(-1/3))^440 + w^20*(T^(-1/3)\
)^476 + w^2*(T^(-1/3))^488 + w^20*(T^(-1/3))^500 + w^14*(T^(-1/3))^1496 + O((T\
^(-1/3))^1498), 
  w^4*(T^(-1/3))^-4 + w^10*(T^(-1/3))^8 + w^4*(T^(-1/3))^20 + w^22*(T^(-1/3))^\
56 + w^4*(T^(-1/3))^68 + w^22*(T^(-1/3))^80 + w^22*(T^(-1/3))^116 + w^4*(T^(-1\
/3))^128 + w^22*(T^(-1/3))^140 + w^4*(T^(-1/3))^176 + w^10*(T^(-1/3))^188 + w^\
4*(T^(-1/3))^200 + w^4*(T^(-1/3))^296 + w^10*(T^(-1/3))^308 + w^4*(T^(-1/3))^3\
20 + w^22*(T^(-1/3))^356 + w^4*(T^(-1/3))^368 + w^22*(T^(-1/3))^380 + w^22*(T^\
(-1/3))^416 + w^4*(T^(-1/3))^428 + w^22*(T^(-1/3))^440 + w^4*(T^(-1/3))^476 + \
w^10*(T^(-1/3))^488 + w^4*(T^(-1/3))^500 + w^22*(T^(-1/3))^1496 + O((T^(-1/3))\
^1498) ]

</pre>

<p>


<hr>

<p>
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