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 [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].
Highlights
We study the Diophantine problem in different classes of finitely generated solvable groups, 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 that the Diophantine problem in O is polynomial time reducible to the Diophantine problem in G
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 that the Diophantine problem in O is effectively polynomial time reducible to the Diophantine problem in G
Summary
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. The ring Z is e-interpretable in G, and D(G) is undecidable Combining this with a result from [14] we obtain that if G is a finitely generated non-virtually abelian nilpotent group of class 2 with infinite cyclic commutator subgroup, D(G) is undecidable, while single equations are decidable in G. If G contains a non-virtually abelian nilpotent subgroup there exists a ring of algebraic integers O e-interpretable in G, D(O) ≤P D(G) This result allows, in particular, to extend Theorem 4.6 to the class of finitely generated virtually nilpotent groups (since the class Nmax is closed under finite extensions).
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