Power-saving bounds for Thue--Mahler and Mordell equations

We prove a new effective bound for cubic Thue--Mahler equations with power-saving dependence on the regulator. This has some applications. First, we improve Stark's bound $\log \max\{|x|,|y|\} \ll_ε|k|^{1+ε}$ for the integer solutions of Mordell's equation $y^2=x^3+k$ ($k$ a non-zero integer) by red