Zeros of linear recurrence sequences

Let f(x) = Pg(x)alpha(0)(x) f + Pk(x)alpha(k)(x) be an exponential polynomi al over a field of zero characteristic. Assume that for each pair i, j with i not equal j, alpha(i) /alpha(j) is not a root of unity. Define Delta = S igma(j=0)(k)(deg P-j + 1). We introduce a partition of {alpha(0),...,alpha( k)} into subsets {alpha(io),...,alpha(iki)} (1 less than or equal to i less than or equal to m), which induces a decomposition of f into f = f(1) + f(m) so that, for 1 less than or equal to i less than or equal to m, (alpha (i0):...:alpha(iki)) is an element of P-ki((Q) over bar) while for 1 less t han or equal to i not equal u less than or equal to m, the number alpha(i0) /alpha(u0) either is transcendental or else is algebraic with not too small a height. Then we show that for all but at most exp(Delta(5 Delta)(5 Delta )) solutions x is an element of Z of f(x) = 0, we have f(1)(x) = ... = f(m)(x)= 0.