  We introduce a subexponential algorithm for geometric solving of multivariate
polynomial equation systems whose bit complexity depends mainly on intrinsic
geometric invariants of the solution set. From this algorithm, we derive a new
procedure for the decision of consistency of polynomial equation systems whose
bit complexity is subexponential, too. As a byproduct, we analyze the division
of a polynomial modulo a reduced complete intersection ideal and from this, we
obtain an intrinsic lower bound for the logarithmic height of diophantine
approximations to a given solution of a zero--dimensional polynomial equation
system. This result represents a multivariate version of Liouville's classical
theorem on approximation of algebraic numbers by rationals. A special feature
of our procedures is their {\em polynomial} character with respect to the
mentioned geometric invariants when instead of bit operations only arithmetic
operations are counted at unit cost. Technically our paper relies on the use of
straight--line programs as a data structure for the encoding of polynomials, on
a new symbolic application of Newton's algorithm to the Implicit Function
Theorem and on a special, basis independent trace formula for affine Gorenstein
algebras.
