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?
  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}}. If Matiyasevich's conjecture on finite-fold Diophantine representations is true, then for every computable function f:N->N there is a positive integer m(f) such that for each integer n>=m(f) there exists a system S \subseteq E_n which has at least f(n) and at most finitely many solutions in integers x_1,...,x_n. This conclusion contradicts to the author's conjecture on integer arithmetic, which implies that the heights of integer solutions to a Diophantine equation are computably bounded, if these solutions form a finite set. 
Apoloniusz Tyszka
math.LO math.NT