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Preprint Number 1426

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1426. Albert Garreta and Alexei Miasnikov and Denis Ovchinnikov
Diophantine problems in solvable groups
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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

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Full text arXiv 1805.04085: pdf, ps.


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