 Authors
 Year
 2002
 Title
 Multivariate Diophantine equations with many solutions
 Publisher
 [S.n.]
 Document type
 Working paper
 Faculty
 Faculty of Science (FNWI)
 Institute
 Kortewegde Vries Institute for Mathematics (KdVI)
 Abstract

Among other things we show that for each ntuple 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 Sunits 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 Sunit 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 equationN_{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.
 Permalink
 http://hdl.handle.net/11245/1.418826
 Downloads
Disclaimer/Complaints regulations
If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library, or send a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible.