Multivariate Diophantine equations with many solutions

Among other things we show that for each n-tuple of positive rational numbers (a_1,..., a_n) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a_1x_1+..+a_nx_n=1 with x_1,..,x_n S-units are notcontained in fewer than exp((4+o(1)) s^{1/2} (log s)^{-1/2} proper linear subspaces of C^n. This generalizes a result of Erdos, Stewart and Tijdeman for S-unit equations in two variables. Further, we prove that for any algebraic number field K of degree n, any integer m with 1<= m<n, and any sufficiently large s there are integers b_0,..,b_min K which are linearly independent over the rationals, and prime numbers p_1,..,p_s, such that the normpolynomial equation|N_{K/\Q}(b_0+b_1x_1+.. +b_mx_m)|=p_1^{z_1}..p_s^{z_s} has at leastexp(1+o(1)){n/m}s^{m/n}(\log s)^{-1+m/n}solutions in integers x_1,..,x_m,z_1,..,z_s.This generalizes a result of Moree and Stewart for m=1. Our main tool, also established in this paper, is an effective lower bound for the number of ideals in a number field K of norm <=X composed of prime ideals which lie outside a given finite set of prime ideals T and which have norm <=Y. This generalizes results of Canfield, Erdos and Pomerance and of Moree and Stewart