On the abc$abc$ conjecture in algebraic number fields.

In this paper, we prove a weak form of the abc$abc$ conjecture generalised to algebraic number fields. Given integers satisfying a+b=c$a+b=c$, Stewart and Yu were able to give an exponential bound in terms of the radical over the integers (Stewart and Yu [Math. Ann. 291 (1991), 225–230], Stewart and Yu [Duke Math. J. 108 (2001), no. 1, 169–181]), whereas Győry was able to give an exponential bound in the algebraic number field case for the projective height HK(a,b,c)$H_{K}(a,\,b,\,c)$ in terms of the radical for algebraic numbers (Győry [Acta Arith. 133 (2008), 281–295]). We generalise Stewart and Yu's method to give an improvement on Győry's bound for algebraic integers over the Hilbert Class Field of the initial number field K. Given algebraic integers a,b,c$a,\,b,\,c$ in a number field K satisfying a+b=c$a+b=c$, we give an upper bound for the logarithm of the projective height HL(a,b,c)$H_{L}(a,\,b,\,c)$ in terms of norms of prime ideals dividing abcOL$abc \mathcal {O}_{L}$, where L is the Hilbert Class Field of K. In many cases, this allows us to give a bound in terms of the modified radical G:=G(a,b,c)$G:=G(a,\,b,\,c)$ as given by Masser (Proc. Amer. Math. Soc. 130 (2002), no. 11, 3141–3150). Furthermore, by employing a recent bound of Győry (Publ. Math. Debrecen 94 (2019), 507–526) on the solutions of S‐unit equations, our estimates imply the upper bound logHLa,b,c<G13+ClogloglogGloglogG,$$\begin{equation*} \hspace*{4pc}\log H_{L}{\left(a,\,b,\,c \right)}< G^{\frac{1}{3}+\mathcal {C}\frac{\log \log \log G}{\log \log G}}, \end{equation*}$$where C$\mathcal {C}$ is an effectively computable constant. Further, given conditions on the largest prime ideal dividing abcOL$abc \mathcal {O}_{L}$, we obtain a sub‐exponential bound for HL(a,b,c)$H_{L}(a,\,b,\,c)$ in terms of the radical. Independently, as a direct application of his bounds on the solutions of S‐unit equations(Győry ([Publ. Math. Debrecen 94 (2019), 507–526]), Győry (Publ. Math. Debrecen 100 (2022), 499–511) also attains results mentioned above, including the above inequality, but over the base field K, as discussed in Section 6. As a consequence of our results, we will give an application to the effective Skolem–Mahler–Lech problem and give an improvement to a result by Lagarias and Soundararajan (J. Théor. Nombres Bordeaux 23 (2011), no. 1, 209–234) on the XYZ conjecture. [ABSTRACT FROM AUTHOR]