Diophantine problems in solvable groups
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 GG in one of these classes, we prove that there exists a ring of algebraic integers OO that is interpretable in GG by finite systems of equations (ee-interpretable), and hence that the Diophantine problem in OO is polynomial time reducible to the Diophantine problem in GG. One of the major open conjectures in number theory states that the Diophantine problem in any such OO is undecidable. If true this would imply that the Diophantine problem in any such GG is also undecidable. Furthermore, we show that for many particular groups GG as above, the ring OO is isomorphic to the ring of integers 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,ℤ),n≥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,ℤ),SL(3,ℤ),T(3,ℤ).
Communicated by Efim Zelmanov