TY - JOUR
T1 - The Diophantine problem in the classical matrix groups
AU - Myasnikov, A. G.
AU - Sohrabi, M.
N1 - Publisher Copyright:
© 2021 Russian Academy of Sciences (DoM) and London Mathematical Society.
PY - 2021/11
Y1 - 2021/11
N2 - In this paper we study the Diophantine problem in the classical matrix groups GLn(R), SLn(R), Tn(R) and UTn(R), n ≥ 3, over an associative ring R with identity. We show that if Gn(R) is one of these groups, then the Diophantine problem in Gn(R) is polynomial-time equivalent (more precisely, Karp equivalent) to the Diophantine problem in R. When Gn(R) = SLn(R) we assume that R is commutative. Similar results hold for PGLn(R) and PSLn(R) provided R has no zero divisors (for PGLn(R) the ring R is not assumed to be commutative).
AB - In this paper we study the Diophantine problem in the classical matrix groups GLn(R), SLn(R), Tn(R) and UTn(R), n ≥ 3, over an associative ring R with identity. We show that if Gn(R) is one of these groups, then the Diophantine problem in Gn(R) is polynomial-time equivalent (more precisely, Karp equivalent) to the Diophantine problem in R. When Gn(R) = SLn(R) we assume that R is commutative. Similar results hold for PGLn(R) and PSLn(R) provided R has no zero divisors (for PGLn(R) the ring R is not assumed to be commutative).
KW - Diophantine problems
KW - classical matrix groups
KW - decidability
KW - equations
KW - undecidability
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U2 - 10.1070/IM9104
DO - 10.1070/IM9104
M3 - Article
AN - SCOPUS:85124245374
SN - 1064-5632
VL - 85
SP - 1220
EP - 1256
JO - Izvestiya Mathematics
JF - Izvestiya Mathematics
IS - 6
ER -