On the solvability of a class of diophantine equations and applications

Abstract

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.