Explicit and mixed estimates for Thue inequalities with few coefficients

Let $F(x,y)$ be an irreducible form of degree $r\geq3$, having $s+1$ non-zero coefficients. Let $h\geq1$ 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.