On the solvability of a class of diophantine equations and applications

For 1 ≤ i ≤ k, let Ri denote pi(y)Fi + Gi, where pi(y) is a polynomial in y with integer coefficients, and Fi, Gi are linear polynomials in x1,..., xn with integer coefficients. Let P(z1,..., zk) be a Presburger relation over the nonnegative integers. We show that the following problem is decidable:Given: R1,..., Rk and a Presburger relation P.Question: Are there nonnegative integer values for y, x1,..., xn such that for these values, (R1,..., Rk) satisfies P? We also give some applications to decision problems concerning counter machines.