Decomposable form equations

S-unit equations and their applications

“…It is a remarkable fact that this bound is independent of the coefficients of F. As a consequence of Theorem 14, Evertse and Gyory [22] derived also explicit bounds for the numbers of solutions of (11.5) and (11.6), with similar conditions as for the effective results.…”

“…Theorem 14 (Evertse and Gyory [22]). With the above notation and under assumption (e), the number of solutions of (11.7) is at most n( 4 x 72g(d+s+w(/3))r-1.…”

“…Here and elsewhere C is a constant, the value of which may be different at each occurrence. Later, Evertse and Gyory [22] derived an upper bound for the number of solutions of (1.8) independent of a 1 and a2 in the general case that the variables x, y belong to a finitely generated multiplicative subgroup of C*. The dependence of Evertse's bound on the degree of K and the cardinality of S is necessary.…”

“…Finally, we note that Theorems 14, 15 were proved in [22], [24], respectively, in the more general form when the ground ring is an arbitrary finitely generated extension ring of Z. In the proofs the authors used Theorem l' and the general version of Theorem 2, respectively.…”

“…By means of (a generalisation of) Theorem 2, Evertse (19] and later Evertse and Gyory [22] derived explicit upper bounds for the numbers of solutions of (11.1) and (11.1') which are independent of the coefficients of F. In [22], the bound 4n X 729(d+s+w(p)) has been obtained for the number of solutions of (11.l') where n = deg(F), g = [G: K] (hence 1 ~ g ~ n!) and w(/3) denotes the number of distinct prime ideal divisors of (/3).…”