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. © 2006 Elsevier B.V. All rights reserved.