Diophantine inequalities involving several power sums

摘要

Let epsilon(A) denote the ring of power sums, i.e. complex functions of the formG(n) = a(1)alpha(1)(n) 1 + a(2)alpha(2)(n) +... + a(t)alpha(t)(n),for some a(i) is an element of C and alpha(i) is an element of A, where A subset of or equal to C is a multiplicative semigroup. Moreover, let F(n, y) is an element of epsilon(A)[y]. We consider Diophantine inequalities of the formF(n, y) < α(n(d- 1- ε)),where α > 1 is a quantity depending on the dominant roots of the power sums appearing as coefficients in F( n, y), and show that all its solutions (n, y) is an element of N x Z have y parametrized by some power sums from a finite set.This is a continuation of the work of Corvaja and Zannier [ 4 - 6] and of the authors [10, 18] on such problems.