0 Ju l 2 00 1 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 1 x 1 +· · ·+a n x n = 1 with x 1 ,. .. , x n S-units are not contained 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 Erd˝ os, Stewart and Tijdeman [7] 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 α 0 ,. .. , α m in K which are linearly independent over Q, and prime numbers p 1 ,. .. , p s , such that the norm polynomial equation |N K/Q (α 0 + α 1 x 1 + · · · + α m x m)| = p z 1 1 · · · p zs s has at least exp{(1+o(1)) n m s m/n (log s) −1+m/n } solutions in x z s ∈ Z. This generalizes a result of Moree and Stewart [19] for m = 1. Our main tool, also established in this paper, is an effective lower bound for the number ψ K,T (X, Y) 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, Erd˝ os and Pomerance [6] and of Moree and Stewart [19].