  In this paper, we study iterative methods on the coefficients of the rational
univariate representation (RUR) of a given algebraic set, called global Newton
iteration. We compare two natural approaches to define locally quadratically
convergent iterations: the first one involves Newton iteration applied to the
approximate roots individually and then interpolation to find the RUR of these
approximate roots; the second one considers the coefficients in the exact RUR
as zeroes of a high dimensional map defined by polynomial reduction, and
applies Newton iteration on this map. We prove that over fields with a p-adic
valuation these two approaches give the same iteration function, but over
fields equipped with the usual Archimedean absolute value, they are not
equivalent. In the latter case, we give explicitly the iteration function for
both approaches. Finally, we analyze the parallel complexity of the different
versions of the global Newton iteration, compare them, and demonstrate that
they can be efficiently computed. The motivation for this study comes from the
certification of approximate roots of overdetermined and singular polynomial
systems via the recovery of an exact RUR from approximate numerical data.
