On the abc conjecture in algebraic number fields

In this paper, we prove a weak form of the abc conjecture generalised to algebraic number fields. Given integers satisfying 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 Gyory was able to give an exponential bound in the algebraic number field case for the projective height H-k (a, b, c) in terms of the radical for algebraic numbers (Gyory [Acta Arith. 133 (2008), 281-295]). We generalise Stewart and Yu's method to give an improvement on Gyory's bound for algebraic integers over the Hilbert Class Field of the initial number field K. Given algebraic integers a, b, c in a number field.. satisfying a + b = c, we give an upper bound for the log-arithm of the projective height H-L(a, b, c) in terms of norms of prime ideals dividing abcO(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) as given by Masser (Proc. Amer. Math. Soc. 130 (2002), no. 11, 3141-3150). Furthermore, by employing a recent bound of Gyory (Publ. Math. Debrecen 94 (2019), 507-526) on the solutions of S-unit equations, our estimates imply the upper bound log H-L (a, b, c) < G(1/3+Clog log log G/log log G), where C is an effectively computable constant. Further, given conditions on the largest prime ideal dividing abcO(L), we obtain a sub-exponential bound for 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(Gyory ([Publ. Math. Debrecen 94 (2019), 507-526]), Gyory (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. Theor. Nombres Bordeaux 23 (2011), no. 1, 209-234) on the XYZ conjecture.