An NP-complete number-theoretic problem

Systems of nonlinear equations of the form D: Ay = σ.(x), where A is an m×n matrix of rational constants and y = (Y1,...,yn), σ(x) = (σ1(x),..., σm (x)) are column vectors are considered. Each σi(x) is of the form ri(x) or @@@@ri(x)@@@@, where ri(x) is a rational function of x with rational coefficients. It is shown that the problem of determining for a given system D whether there exists a nonnegative integral solution (y1,...,yn,X) satisfying D is decidable. In fact, the problem is NP-complete when restricted to systems D in which the maximum degree of the polynomials defining the σi(x)'s is bounded by some fixed polynomial in the length of the representation of D. Some recent results connecting Diophantine equations and counter machines are briefly mentioned.