Number Theory: An Introduction to Mathematics

7 Complements 215

where‖x‖=max(|x 1 |,...,|xn|), is contained in some subspaceVi, except for finitely
many points whose number may depend onε. A new proof of Schmidt’s subspace the-
orem has been given by Faltings and W ̈ustholz (1994). The subspace theorem has also
been given a more quantitative form by Schmidt (1989) and Evertse (1996). These
results have immediate applications to the simultaneous approximation of several
algebraic numbers.
Vojta (1987) has developed a remarkable analogy between the approximation of
algebraic numbers by rationals and the theory of Nevanlinna (1925) on the value dis-
tribution of meromorphic functions, in which Roth’s theorem corresponds to Nevan-
linna’s second main theorem. Although the analogy is largely formal, it is suggestive in
both directions. It has already led to new proofs for the theorems of Roth and Schmidt,
and to a proof of the Mordell conjecture (discussed below) which is quite different
from the original proof by Faltings.
Roth’s theorem has an interesting application to Diophantine equations. Let

f(z)=a 0 zn+a 1 zn−^1 +···+an

be a polynomial of degreen≥3 with integer coefficients whose roots are distinct and
not rational. Let

f(u,v)=a 0 un+a 1 un−^1 v+···+anvn

be the corresponding homogeneous polynomial and letg(u,v)be a polynomial of
degreem≥0 with integer coefficients. We will deduce from Roth’s theorem that the
equation

f(x,y)=g(x,y)

has at most finitely many solutions in integers ifm≤n−3. This was already proved
by Thue form=0.
Assume on the contrary that there exist infinitely many solutions in integers. With-
out loss of generality we may assume that there exist infinitely many integer solutions
x,yfor which|x|≤|y|. Then there exists a constantc 1 >0 such that

|g(x,y)|≤c 1 |y|m.

Over the complex fieldCthe homogeneous polynomialf(u,v)has a factorization

f(u,v)=a 0

∏n

j= 1

(u−ζjv),

whereζ 1 ,...,ζnare distinct algebraic numbers which are not rational. For at least one
jwe must have, for infinitely manyx,y,

|a 0 ||x−ζjy|n≤c 1 |y|m

and hence

|x−ζjy|≤c 2 |y|m/n,