Submission date: 10 May 2018

Abstract:

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 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 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.

Mathematics Subject Classification: 20F70, 20F10, 03B25, 03D35, 20F18, 20F16

Keywords and phrases:

Full text arXiv 1805.04085: pdf, ps.