This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFT-based power-series exponentiation; explains how one can choose the parameters to achieve any desired level of accuracy; and discusses several generalizations.

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 eld K of degree n, any integer m with 1 m < n, and any suciently large s there are integers 0 ; : : : ; m in K which are linearly independent over Q , and prime numbers p 1 ; : : : ; p s , such that the norm polynomial equation jN K=Q ( 0 + 1 x 1 + + mxm )j = p z1 1 p zs s has at least expf(1+o(1)) n m s m=n (log s) 1+m=n g solutions in x 1 ; : : : ; xm ; z 1 ; : : : ; z s 2 Z. This generalizes a result of Moree and Stewart [19] for m = 1. Our main tool, also established in this paper, is an eective lower bound for the number K;T (X; Y ) of ideals in a number eld K of norm X composed of prime ideals which lie outside a given nite set of prime ideals T and which have norm Y . This generalizes results of Caneld, Erd}os and Pomerance [6] and of Moree and Stewart [19]. 2000 Mathematics Subject Classication: 11D57, 11D61. The research of the third author was supported in part by Grant A3528 from the Natural Sciences and Engineering Research Council of Canada. 1 1