  Newton's method is used to approximate roots of complex valued functions f by
creating a sequence of points that converges to a root of f in the usual
topology. For any field K equipped with a set of pairwise inequivalent absolute
values satisfying a product formula, we completely describe the conditions
under which Newton's method applied to a squarefree polynomial f with
K-coefficients will succeed in finding a root of f in the v-adic topology for
infinitely many places v of K. Furthermore, we show that if K is a finite
extension of the rationals or of the rational function field over a finite
field, then the Newton approximation sequence fails to converge v-adically for
a positive density of places v.
