Multivariate Diophantine equations with many solutions

Details Multivariate Diophantine equations with many solutions Multivariate Diophantine equations with many solutions

2001-07-30

arxiv.org/abs/math/0107219v1

Acta Arith. 107 (2003), 103-125

Mathematics

Number Theory

29 pages

Scientific paper

We show that for each n-tuple of positive rational integers (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 the x_i all 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 Erdos, Stewart and Tijdeman for m=2 [Compositio 36 (1988), 37-56]. Furthermore we prove that for any algebraic number field K of degree n, any integer m with 1<=m

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