The abc-conjecture for a number field

Let $K$ be a number field. For $x\in K^\times$, define its height by $\operatorname{h}(x)$ and its conductor by

$$\operatorname{cond}(x)=d^{-1}\sum_{|x|_v<1}\log\mathcal{N}v.$$

The abc-conjecture for a number field. For every $\kappa>1$ and every $x\in K$ distinct from $0$ and $1$,

$$\operatorname{h}(x)\le \kappa\bigl(\operatorname{cond}(x)+\operatorname{cond}(x^{-1})+\operatorname{cond}(x-1)\bigr)+O(1),$$

where $O(1)$ depends on $\kappa$ and $K$. This is the number-field form of the abc-conjecture, originally stated in this form by Vojta and related to the Masser–Oesterlé conjecture.