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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 finite-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 S \subseteq E_n which hasat least f(n) and at most finitely many solutions in integers x_1,...,x_n. Thisconclusion contradicts to the author's conjecture on integer arithmetic, whichimplies that the heights of integer solutions to a Diophantine equation arecomputably bounded, if these solutions form a finite set.

Does there exist an algorithm which to each Diophantine equation assigns  an integer which is greater than the number (heights) of integer solutions,  if these solutions form a finite set?