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 [Formula: see text] in one of these classes, we prove that there exists a ring of algebraic integers [Formula: see text] that is interpretable in [Formula: see text] by finite systems of equations ([Formula: see text]-interpretable), and hence that the Diophantine problem in [Formula: see text] is polynomial time reducible to the Diophantine problem in [Formula: see text]. One of the major open conjectures in number theory states that the Diophantine problem in any such [Formula: see text] is undecidable. If true this would imply that the Diophantine problem in any such [Formula: see text] is also undecidable. Furthermore, we show that for many particular groups [Formula: see text] as above, the ring [Formula: see text] is isomorphic to the ring of integers [Formula: see text], so the Diophantine problem in [Formula: see text] 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 [Formula: see text]. 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 [Formula: see text].
Polynomial time conjugacy in wreath products and free solvable groups