Let $F : R^k \to R^k$ be a smooth map whose Jacobian
$J(F): R^k \to R$ vanishes on a discrete set $S$, so that if
$O$ is the complement of $S$, then $f: O \to R^k$, the
restriction of $F$, is a local diffeomorphism, and in particular
there are only finitely many points $x$ inside the unit ball of
$R^k$ such that $f(x) = 0$. What I would like to know is if
there any a good algorithm for finding such points $x$.
Of course, if $k=1$ this is easy, using the Intermediate Value
Theorem; just partition the interval $[-1,1]$ into a reasonably large
number $N$ of subintervals of length $2/N$ and test on which
subintervals $f$ changes sign, and then use bisection to locate
a zero in those subintervals. If $k>1$ I suspect that some sort
of application of Marching Cubes may work, but the usual search
methods have not turned up anything.