Simultaneous Diophantine Approximation in Function Fields

Abstract: 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.