CoCoA » CoCoA-5

@ApproxSolveTF@ is not documented. Maybe that is a good thing...

<pre>
use QQ[x,y,z];

D := 7;
N := 2^17;
L := [z^D - (N*z-1)^2, y*(z-1/N)-1, x^D - (y^2*x-1)^2];
Pts := ApproxSolveTF(L);
indent([[ FloatStr(coord) | coord in pt] | pt in Pts]);
[
["0.0000", "0.0000", "0.0000"],
["0.0000", "0.0000", "0.0000"],
["0.0000", "0.0000", "0.0000"],
["0.0000", "0.0000", "0.0000"],
["0.99998", "0.0089742", "111.43"],
["0.0000", "0.0000", "0.0000"],
["0.0000", "0.0000", "0.0000"]
]
</pre>

I'm not quite sure what happened there. @ApproxSolve@ can handle this (if I increase the internal precision enough): it finds these solutions:
<pre>
[
["8.7581*10^(-47)", "-1.0685*10^23", "0.0000076294"],
["8.7581*10^(-47)", "-1.0685*10^23", "0.0000076294"],
["8.7581*10^(-47)", "1.0685*10^23", "0.0000076294"],
["8.7581*10^(-47)", "1.0685*10^23", "0.0000076294"],
["0.99998", "0.0089742", "111.43"],
["2.6487*10^18", "1.0685*10^23", "1.2768*10^262", "0.0000076294"], "1.1182*10^216"],
["2.6487*10^18", "-1.0685*10^23", "4.9626*10^260", "0.0000076294"] "4.3463*10^214"]
]
</pre>