Abstract:
Let F(X, Y ) = s
i=0 ai Xri Yr−ri ∈ Z[X, Y ] be a form of degree
r = rs ≥ 3, irreducible over Q and having at most s + 1 non-zero coefficients. Mueller
and Schmidt showed that the number of solutions of the Thue inequality
|F(X, Y )| ≤ h
is s2h2/r (1 + log h1/r ). They conjectured that s2 may be replaced by s. Let
= max
0≤i≤s
max
⎛
⎝
i−1
w=0
1
ri − rw
,
s
w=i+1
1
rw − ri
⎞
⎠
.
Then we show that s2 may be replaced by max(s log3 s, se ). We also show that if
|a0| = |as | and |ai| ≤ |a0| for 1 ≤ i ≤ s − 1, then s2 may be replaced by s log3/2 s. In
particular, this is true if ai ∈ {−1, 1}.