• Contributions to a conjecture of Mueller and Schmidt on Thue inequalities

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/127/04/0565-0584

• # Keywords

Thue equations; Thue inequalities; Archimedean Newton polygon

• # Abstract

Let $F(X, Y) = \sum^{s}_{i=0}a_{i}X^{r_i}Y^{r−r_i} \in \mathbb{Z}[X, Y]$ be a form of degree $r = r_{s} \geq 3$, 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 $$\mid F(X, Y) \mid \leq h$$ is $\ll s^{2}h^{2/r}(1+log h^{1/r})$. They conjectured that $s^{2}$ may be replaced by $s$. Let $$\Psi = \mathop{\max}\limits_{0\leq i\leq s} max \left(\sum^{i-1}_{w=0} \frac{1}{r_{i}-r_{w}}, \sum^{s}_{w=i+1}\frac{1}{r_{w}-r_{i}}\right)$$. Then we show that $s^2$ may be replaced by ${max(s\log^{3} s, se^{\Psi})}$. We also show that if $\mid{a_0}\mid = \mid{a_s}\mid$ and $\mid{a_i} \leq \mid{a_0}\mid$ for $1 \leq i \leq s − 1$, then $s^2$ may be replaced by $s\log^{3/2} s$. In particular, this is true if $a_{i}\in {−1, 1}$.

• # Author Affiliations

1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019