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?

TL;DR

The paper explores the relationship between finite solutions of Diophantine equations, the heights of solutions, and conjectures in number theory, suggesting that certain assumptions lead to contradictions regarding the computability of solution heights.

Contribution

It connects Matiyasevich's conjecture on finite-fold Diophantine representations with a conjecture on bounded solution heights, highlighting potential conflicts.

Findings

If Matiyasevich's conjecture holds, then solutions can have arbitrarily large heights.

The author's conjecture implies a computable bound on solution heights for finite solutions.

A contradiction arises if both conjectures are assumed true, indicating a tension in current number theory hypotheses.

Abstract

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.