  There are abundant results on Diophantine approximation over fields of
positive characteristic (see the survey papers [13, 25]), but there is very
little information about simultaneous approximation. In this paper, we develop
a technique of \geometry of numbers" in positive characteristic, so that we may
generalize some of the classical results on simultaneous approximation to the
case of function fields. More precisely, we approximate a finite set of Laurent
series by rational functions with a common denominator. In particular, the
lower bound results we obtain may be regarded as a high dimensional version of
the Liouville{ Mahler Theorem on algebraic functions of degree n. As an
application, we investigate binary quadratic forms, and determine the exact
approximation constant of a quadratic algebraic function. Finally, we give two
examples using continued fractions.
