Contributions to a conjecture of Mueller and Schmidt on Thue inequalities

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}.