On polynomial systems of equations in square matrices filled with natural numbers

TL;DR

This paper explores the undecidability of positive existential theories of matrix sets over natural numbers, revealing their complex logical structure and connections to divisibility lattices.

Contribution

It establishes the undecidability of the positive existential theory of matrix sets over natural numbers and relates it to divisibility lattice structures.

Findings

The positive existential theories of $M_n(\mathbb N)$ are undecidable.

These theories form an inclusion lattice isomorphic to the divisibility lattice.

Undecidability extends to Diophantine equations with coefficients in $M_n(\mathbb Z)$.

Abstract

The positive existential theories of the sets $M_n(\mathbb N)$ without parameters build an inclusion lattice isomorhic with the lattice of divisibility. All these sets are algorithmically undecidable. In further sections some easier observations are made, like the undecidability of Diophantine equations with coefficients in $M_n(\mathbb Z)$.