Systems of nonlinear equations of the form D: A&ymarc; &equil; &sgr;marc;.(x), where A is an m×n matrix of rational constants and &ymarc; &equil; (Y<subscrpt>1</subscrpt>,...,y<subscrpt>n</subscrpt>), &sgr;(x) &equil; (&sgr;<subscrpt>1</subscrpt>(x),..., &sgr;<subscrpt>m</subscrpt> (x)) are column vectors are considered. Each &sgr;<subscrpt>i</subscrpt>(x) is of the form r<subscrpt>i</subscrpt>(x) or @@@@r<subscrpt>i</subscrpt>(x)@@@@, where r<subscrpt>i</subscrpt>(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 (y<subscrpt>1</subscrpt>,...,y<subscrpt>n</subscrpt>,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 &sgr;<subscrpt>i</subscrpt>(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.
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