TY - JOUR
T1 - Diophantine problems in solvable groups
AU - Garreta, Albert
AU - Miasnikov, Alexei
AU - Ovchinnikov, Denis
N1 - Publisher Copyright:
© 2020 Lippincott Williams and Wilkins. All rights reserved.
PY - 2020/4/1
Y1 - 2020/4/1
N2 - 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 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 hence that the Diophantine problem in O is polynomial time reducible to the Diophantine problem in G. One of the major open conjectures in number theory states that the Diophantine problem in any such O is undecidable. If true this would imply that the Diophantine problem in any such G is also undecidable. Furthermore, we show that for many particular groups G as above, the ring O 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). ;copy; 2020 The Author(s).
AB - 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 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 hence that the Diophantine problem in O is polynomial time reducible to the Diophantine problem in G. One of the major open conjectures in number theory states that the Diophantine problem in any such O is undecidable. If true this would imply that the Diophantine problem in any such G is also undecidable. Furthermore, we show that for many particular groups G as above, the ring O 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). ;copy; 2020 The Author(s).
KW - Diophantine problem
KW - Hilbert's 10th problem
KW - ring of algebraic integers
KW - solvable groups
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U2 - 10.1142/S1664360720500058
DO - 10.1142/S1664360720500058
M3 - Article
AN - SCOPUS:85081613577
SN - 1664-3607
VL - 10
JO - Bulletin of Mathematical Sciences
JF - Bulletin of Mathematical Sciences
IS - 1
M1 - 2050005
ER -