Studying the Diophantine problem in finitely generated rings and algebras via bilinear maps

TL;DR

This paper explores the decidability of polynomial equations in finitely generated rings and algebras by interpreting these structures through systems of equations, revealing undecidability in many cases and introducing a novel technique involving bilinear maps.

Contribution

It introduces a method to interpret finitely generated rings and algebras via bilinear maps, linking their Diophantine problems to those of well-understood rings, and extends results to various classes of rings and algebras.

Findings

Diophantine problem in certain rings is reducible to that in rings of integers of number fields.

In some cases, the Diophantine problem in these rings is shown to be undecidable.

A new technique using bilinear maps interprets complex structures within systems of equations.

Abstract

We study systems of polynomial equations in several classes of finitely generated rings and algebras. For each ring $R$ (or algebra) in one of these classes 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 Karp-reducible to the same problem in $R$. In several cases we further obtain an interpretation by systems of equations of the ring $\mathbb{F}_p[t]$ in $R$, which implies that the Diophantine problem in $R$ is undecidable in this case. Otherwise, the ring $O$ is a ring of algebraic integers, and then the long-standing conjecture that $\mathbb{Z}$ is always interpretable by systems of equations in $O$ carries over to $R$. If true, it implies that the Diophantine problem in $R$ is also undecidable. Some of the classes of f.g. rings studied in this paper are the following: all associative, commutative, non-unitary rings (a similar statement for the unitary case was obtained by Eisentraeger); all possibly non-associative, non-commutative non-unitary rings that are f.g. as an abelian group; and several classes of f.g. non-commutative rings. Analogous statements are obtained for algebras over f.g. associative commutative unitary rings. Another contribution is the technique by which the aforementioned results are obtained: We show that given a bilinear map $f: A\times B \to C$ between f.g. abelian groups (or modules), under mild assumptions, there exists a certain ring (or algebra) $R$ with nice properties which is interpretable by systems of equations in the multi-sorted structure $(A,B,C;f)$. This result is not only relevant for rings and algebras, but also in other structures such as groups, as demonstrated previously by the authors.