Diophantine problems in solvable groups

TL;DR

This paper investigates the decidability of solving systems of equations in various finitely generated solvable groups, linking it to the undecidability of similar problems in rings of algebraic integers, and establishing undecidability in several specific groups.

Contribution

It proves that the Diophantine problem in many solvable groups can be reduced to that in rings of algebraic integers, and identifies cases where this problem is undecidable.

Findings

Diophantine problem in certain groups is reducible to that in rings of algebraic integers.

In many groups, the ring of integers is isomorphic to Z, making the problem undecidable.

Undecidability established for groups like $GL(3,bZ)$, $SL(3,bZ)$, and $T(3,bZ)$.

Abstract

We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc), which satisfy some natural "non-commutativity" conditions. For each group $G$ in one of these classes, we prove that there exists a ring of algebraic integers $O$ that is interpretable in $G$ by finite systems of equations (e-interpretable), and hence that the Diophantine problem in $O$ is polynomial time reducible to the Diophantine problem in $G$. One of the major open conjectures in number theory states that the Diophantine problem in any such $O$ is undecidable. If true this would imply that the Diophantine problem in any such $G$ is also undecidable. Furthermore, we show that for many particular groups $G$ as above, the ring $O$ is isomorphic to the ring of integers $\mathbb{Z}$, so the Diophantine problem in $G$ is, indeed, undecidable. This holds, in particular, for free nilpotent or free solvable non-abelian groups, as well as for non-abelian generalized Heisenberg groups and uni-triangular groups $UT(n,\mathbb{Z}), n \geq 3$. Then we apply these results to non-solvable groups that contain non-virtually abelian maximal finitely generated nilpotent subgroups. For instance, we show that the Diophantine problem is undecidable in the groups $GL(3,\mathbb{Z}), SL(3,\mathbb{Z}), T(3,\mathbb{Z})$.