Contributions to a conjecture of Mueller and Schmidt on Thue inequalities

Abstract

Let $F(X,Y)=\sum _{i=0}^s{a}_i{X}^{r_i}{Y}^{r-r_i}\in \mathbb{Z}[X,Y]$ be a form of degree $r=r_s\ge 3$, irreducible over $\mathbb{Q}$ and having at most $s+1$ non-zero coefficients. Mueller and Schmidt showed that the number of solutions of the Thue inequality \[ |F(X,Y)|\leq h \] is $\ll s^2{h}^{2/r}(1+\log h^{1/r})$. They $\text{conjectured}$ that $s^2$ may be replaced by $s$. Let \[ Ψ= \max_{0\leq i\leq s} \max\left( \sum_{w=0}^{i-1}\frac{1}{r_i-r_w},\sum_{w= i+1}^{s}\frac{1}{r_w-r_i}\right). \] Then we show that $s^2$ may be replaced by $\max (s{\log }^{3}s,s{e}^{\Psi })$. We also show that if $|a_0|=|a_s|$ and $|a_i|\le |a_0|$ for $1\le i\le s-1$, then $s^2$ may be replaced by $s{\log }^{3/2}s$. In particular, this is true if $a_i\in \{-1,1\}$.

27 pages, 1 figure