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?

TL;DR

This paper explores the complexity of Diophantine equations, defining a function related to their solutions, and discusses implications for the existence of certain algorithms and representations in number theory.

Contribution

It establishes properties of the function f(n) related to solutions of specific Diophantine systems and links these properties to the existence of algorithms for solving Diophantine equations.

Findings

f(n) is strictly increasing

No finite-fold Diophantine representation exists for functions bounded below by f(n)

Implications for the existence of an algorithm that bounds solutions of finite Diophantine equations

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