How to solve this non-linear system of equations?

[parisse]

For exact data *and* polynomial system, you can find all solutions in theory, therefore conditions on variables *could* be checked. For everything else, there is no hope. The best I can imagine is to compute a grid of points in a given product of intervals, find the smallest in absolute value and start an iterative algorithm with this guess.
This is really something everyone should understand before using fsolve/solve and so. Sorry, no blackbox here...