The Diophantine problem in the classical matrix groups

Abstract

In this paper we study the Diophantine problem in the classical matrix groups , , and , , over an associative ring with identity. We show that if is one of these groups, then the Diophantine problem in is polynomial-time equivalent (more precisely, Karp equivalent) to the Diophantine problem in . When we assume that is commutative. Similar results hold for and provided has no zero divisors (for the ring is not assumed to be commutative).

Regular bi-interpretability of Chevalley groups over local rings

“…A new round of development of this topic has appeared recently in the papers of A. Nies ([37], [36], etc), N. Avni, A. Lubotzky, C. Meiri ( [4], [5], also see [22]), A. Myasnikov, O. Kharlampovich and M. Sohrabi ( [25], [35], [34], etc. ), D. Segal and K. Tent ( [38]), B. Kunyavskii, E. Plotkin, N. Vavilov ( [28]) and others.…”

Regular bi-interpretability of Chevalley groups over local rings

“…Nies ( [49], [48], etc. ), N. Avni, A. Lubotzky, C. Meiri ( [4], [5], also see [32]), A. Myasnikov, O. Kharlampovich and M. Sohrabi ( [34], [45], [46], etc. ), D. Segal and K. Tent ( [55]), B. Kunyavskii, E. Plotkin, N. Vavilov ( [39]) and others appeared (see [52] for a survey of results).…”