On the solvability of a class of diophantine equations and applications

For 1 ⩽ i ⩽ k , let R i denote p i ( y ) F i + G i , where p i ( y ) is a polynomial in y with integer coefficients, and F i , G i are linear polynomials in x 1 , … , x n with integer coefficients. Let P ( z 1 , … , z k ) be a Presburger relation over the nonnegative integers. We show that the following problem is decidable: Given: R 1 , … , R k and a Presburger relation P. Question: Are there nonnegative integer values for y , x 1 , … , x n such that for these values, ( R 1 , … , R k ) satisfies P? We also give some applications to decision problems concerning counter machines