The Diophantine problem in finitely generated commutative rings

TL;DR

This paper investigates the decidability of polynomial equations in finitely generated commutative rings, showing that in many cases the problem is undecidable, and relating it to longstanding conjectures about algebraic integers.

Contribution

It establishes interpretations of rings of integers within finitely generated rings, linking Diophantine problems and undecidability to broader algebraic structures.

Findings

Diophantine problem in rings of positive characteristic is undecidable.

Interpretation of rings of integers within finitely generated rings.

Connection to the conjecture that $ ext{Z}$ is interpretable in rings of algebraic integers.

Abstract

We study systems of polynomial equations in infinite finitely generated commutative associative rings with an identity element. For each such ring $R$ we obtain an interpretation by systems of equations of a ring of integers $O$ of a finite field extension of either $\mathbb{Q}$ or $\mathbb{F}_p(t)$, for some prime $p$ and variable $t$. This implies that the Diophantine problem (decidability of systems of polynomial equations) in $O$ is reducible to the same problem in $R$. If, in particular, $R$ has positive characteristic or, more generally, if $R$ has infinite rank, then we further obtain an interpretation by systems of equations of the ring $\mathbb{F}_p[t]$ in $R$. This implies that the Diophantine problem in $R$ is undecidable in this case. In the remaining case where $R$ has finite rank and zero characteristic, we see that $O$ is a ring of algebraic integers, and then the long-standing conjecture that $\mathbb{Z}$ is always interpretable by systems of equations in a ring of algebraic integers carries over to $R$. If true, it implies that the Diophantine problem in $R$ is also undecidable. Thus, in this case the Diophantine problem in every infinite finitely generated commutative unitary ring is undecidable. The present is the first in a series of papers were we study the Diophantine problem in different types of rings and algebras.