Explicit and Mixed Estimates for Thue inequalities with few coefficients

TL;DR

This paper provides explicit upper bounds for the number of solutions to Thue inequalities involving forms with few non-zero coefficients, improving upon previous bounds by making all constants explicit.

Contribution

The paper introduces three explicit upper bounds for solutions of Thue inequalities with few coefficients, making all constants fully explicit.

Findings

Derived three explicit upper bounds for solutions

Improved upon previous bounds by removing undetermined constants

Applicable to forms with few non-zero coefficients

Abstract

Let $F(x,y)$ be an irreducible form of degree $r\geq 3$ and having $s+1$ non-zero coefficients. Let $h\geq 1$ be an integer and consider the Thue inequality $$|F(x,y)|\leq h.$$ Following the seminal work of Thue in 1909, several papers were written giving an upper bound for the number of solutions of the above inequality as $\ll c(r,s,h)$ where $c(r,s,h)$ is an explicit function of $r,s$ and $h.$ Invariably, the absolute constant involved in $\ll$ has been left undetermined. In this paper, following Bombieri, Schmidt and Mueller, we give three different upper bounds which are explicit in every aspect.