Small systems of Diophantine equations with a prescribed number of solutions in non-negative integers

TL;DR

The paper explores the relationship between Diophantine equations and the number of solutions, proposing that under Matiyasevich's conjecture, systems can be constructed with a prescribed number of solutions in non-negative integers.

Contribution

It links Matiyasevich's conjecture to the ability to construct Diophantine systems with any computable number of solutions, highlighting a potential universality.

Findings

Conditional existence of systems with any computable number of solutions

Connection between Matiyasevich's conjecture and solution counts

Implication for the complexity of Diophantine systems

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 single-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 U \subseteq E_n which has exactly f(n) solutions in non-negative integers x_1,...,x_n.