Diophantine problems in solvable groups

Albert Garreta, Alexei Miasnikov, Denis Ovchinnikov

Published 2018-05-10Version 1

We study systems of equations in different classes of solvable groups. 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 systems of equations (e-interpretable). This leads to the conjecture that $\mathbb{Z}$ is e-interpretable in $G$ and that the Diophantine problem in $G$ is undecidable. %This stems from a long standing conjecture which states the same for the ring $O$. We further prove that $\mathbb{Z}$ is e-interpretable in any generalized Heisenberg group and in any finitely generated nonabelian free (solvable-by-nilpotent) group. The latter applies in particular to the case of free solvable groups and to the already known case of free nilpotent groups.