Geodesic
The Diophantine problem in the classical matrix groups
Izvestiya. Mathematics , Tome 85 (2021) no. 6, pp. 1220-1256

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In this paper we study the Diophantine problem in the classical matrix groups $\mathrm{GL}_n(R)$, $\mathrm{SL}_n(R)$, $\mathrm{T}_n(R)$ and $\mathrm{UT}_n(R)$, $n\geqslant 3$, over an associative ring $R$ with identity. We show that if $G_n(R)$ is one of these groups, then the Diophantine problem in $G_n(R)$ is polynomial-time equivalent (more precisely, Karp equivalent) to the Diophantine problem in $R$. When $G_n(R)=\mathrm{SL}_n(R)$ we assume that $R$ is commutative. Similar results hold for $\mathrm{PGL}_n(R)$ and $\mathrm{PSL}_n(R)$ provided $R$ has no zero divisors (for $\mathrm{PGL}_n(R)$ the ring $R$ is not assumed to be commutative).
Keywords: Diophantine problems, classical matrix groups, decidability, undecidability.
Mots-clés : equations 
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     author = {A. G. Myasnikov and M. Sohrabi},
     title = {The {Diophantine} problem in the classical matrix groups},
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     pages = {1220--1256},
     publisher = {mathdoc},
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     number = {6},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2021_85_6_a6/}
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A. G. Myasnikov; M. Sohrabi. The Diophantine problem in the classical matrix groups. Izvestiya. Mathematics , Tome 85 (2021) no. 6, pp. 1220-1256. http://geodesic.mathdoc.fr/item/IM2_2021_85_6_a6/