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Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. IfMatiyasevich's conjecture on single-fold Diophantine representations is true,then for every computable function f:N->N there is a positive integer m(f) suchthat for each integer n>=m(f) there exists a system U \subseteq E_n which hasexactly f(n) solutions in non-negative integers x_1,...,x_n.

Small systems of Diophantine equations with a prescribed number of solutions in non-negative integers