TY - JOUR
T1 - The Diophantine problem in Chevalley groups
AU - Bunina, Elena
AU - Myasnikov, Alexei
AU - Plotkin, Eugene
N1 - Publisher Copyright:
© 2024
PY - 2024/7/15
Y1 - 2024/7/15
N2 - In this paper we study the Diophantine problem in Chevalley groups Gπ(Φ,R), where Φ is a reduced irreducible root system of rank >1, R is an arbitrary commutative ring with 1. We establish a variant of double centralizer theorem for elementary unipotents xα(1). This theorem is valid for arbitrary commutative rings with 1. The result is principal to show that any one-parametric subgroup Xα, α∈Φ, is Diophantine in G. Then we prove that the Diophantine problem in Gπ(Φ,R) is polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine problem in R. This fact gives rise to a number of model-theoretic corollaries for specific types of rings.
AB - In this paper we study the Diophantine problem in Chevalley groups Gπ(Φ,R), where Φ is a reduced irreducible root system of rank >1, R is an arbitrary commutative ring with 1. We establish a variant of double centralizer theorem for elementary unipotents xα(1). This theorem is valid for arbitrary commutative rings with 1. The result is principal to show that any one-parametric subgroup Xα, α∈Φ, is Diophantine in G. Then we prove that the Diophantine problem in Gπ(Φ,R) is polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine problem in R. This fact gives rise to a number of model-theoretic corollaries for specific types of rings.
KW - Chevalley groups
KW - Diophantine problem
KW - Diophantine set
KW - Double centralizer theorem
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U2 - 10.1016/j.jalgebra.2024.04.003
DO - 10.1016/j.jalgebra.2024.04.003
M3 - Article
AN - SCOPUS:85190475619
SN - 0021-8693
VL - 650
SP - 219
EP - 274
JO - Journal of Algebra
JF - Journal of Algebra
ER -