$S$-unit equations and Masser's $ABC$ conjecture in algebraic number fields

Let $K$ be an algebraic number field, and $S$ a finite set of places on $K$ which contains all infinite places. In terms of $S$, the best known upper bound for the heights of the solutions of the $S$-unit equation (2.1) over $K$ is given in Gy\H ory [21]; see also (2.4) in Section 2 below. In Section 4, we apply this bound to derive the best Masser's type $ABC$ inequalities to date towards Masser's $ABC$ conjecture over $K$; cf. Theorems 1 and 2. Independently, using a different approach, Scoones [31] proves in fact the same theorems but in a slightly weaker form, over the Hilbert class field of $K$ and not over $K$. See also the Remarks in Section 1.
In the opposite direction, in Section 5, we deduce from the effective version of Masser's $ABC$ conjecture over $K$ a significant, but conditional and not completely explicit improvement of the bound (2.4).