## Abstract

In this paper we study the Diophantine problem in Chevalley groups G_{π}(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 x_{α}(1). This theorem is valid for arbitrary commutative rings with 1. The result is principal to show that any one-parametric subgroup X_{α}, α∈Φ, is Diophantine in G. Then we prove that the Diophantine problem in G_{π}(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.

Original language | English |
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Pages (from-to) | 219-274 |

Number of pages | 56 |

Journal | Journal of Algebra |

Volume | 650 |

DOIs | |

State | Published - 15 Jul 2024 |

### Bibliographical note

Publisher Copyright:© 2024

## Keywords

- Chevalley groups
- Diophantine problem
- Diophantine set
- Double centralizer theorem