In this section we will be concerned about the approximation of
functions, algebraic over a global field K of positive characteristic by
elements of K with respect to a valuation v of K.
We define, for 
(although we will consider only y
algebraic over K in what follows):
where 
, where k is the constant field of K.
We will give some examples that exhibit pathological behaviour. Recall
that 
, which are analogues of
the classical theorems of Dirichlet and Liouville.
Osgood [O] has shown that 
if y does not
satisfy a Riccati equation and we can prove the same bound
if the cross ratio of any four conjugates of y over K is non
constant. There are some results on 
if y
satisfies 
where 
and
q is a power of p,
due to the author [V1] and others([BS],[dM],[MR]). One may conjecture that
these are actually the only functions not satisfying Osgood's bound.
We shall give examples that show that Osgood's bound is close to being best
possible.
Take 
and y satisfying 
and 
(y is a
classical example
of Mahler's). We have 
. Also, whenever
is near p we have 
near 
. It
follows that 
. Note that z
does not satisfy a Riccati equation. This example can be generalized as
follows: Given y and 
a rational function of degree d
in Y, then 
and 
. So if
is large we get new examples of well approximated functions which
in general do not satisfy Riccati equations. For other examples and a
proof of the following theorem, see [V6].
By definition, the cross ratio of 
is
Remark: Let D be the divisor on 
formed by the conjugates
of y over K, so D is of degree d and is defined over K. Let X be
the affine curve 
. It can be checked that y satisfies
a Riccati equation if and only if the Kodaira-Spencer class of X, in the
sense of the appendix, vanishes. It can also be checked
that the cross ratio of any four
conjugates of y lies in k if and only if X is isotrivial, that is,
isomorphic to an affine curve defined over k perhaps after field extension.
It then follows from theorem 6 that, when X is non-isotrivial, it has only
finitely many integral points, which gives another proof of theorem 1 in
the genus zero case.
Wang [W] has some results on diophantine approximation in 
for
n > 1. In particular, she has a sufficient condition for finiteness
of the set of integral points of 
minus 2n+2 hyperplanes.
