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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}}. For apositive integer n, let f(n) denote the greatest finite total number ofsolutions of a subsystem of E_n in integers x_1,...,x_n. We prove: (1) thefunction f is strictly increasing, (2) if a non-decreasing function g frompositive integers to positive integers satisfies f(n) \geq g(n) for any n, thena finite-fold Diophantine representation of g does not exist, (3) if thequestion of the title has a positive answer, then there is a computablestrictly increasing function g from positive integers to positive integers suchthat f(n) \leq g(n) for any n and a finite-fold Diophantine representation of gdoes not exist.

Is there an algorithm which takes as input a Diophantine equation,  returns an integer, and this integer is greater than the number of integer  solutions, if the solution set is finite?