Bounded reduction of orthogonal matrices over polynomial rings

Gvozdevsky, Pavel

doi:10.1016/j.jalgebra.2022.02.026

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Regular Bi-interpretability1

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Cited by 3 publications

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“…• In papers [65], [25], [26] it is proved that an element from the Chevalley group of type Φ, where rk Φ 2 and Φ = F 4 , G 2 , E 8 , over a polynomial ring with coefficients in a small-dimensional ring can be reduced to an element of certain proper subsystem subgroup by a bounded number of elementary root elements (the bound depends on the number of variables). By induction, for the ring Z[x 1 , .…”

Section: Regular Bi-interpretabilitymentioning

confidence: 99%

Regular bi-interpretability of Chevalley groups over local rings

Bunina

2022

Preprint

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In this paper we prove that if $G(R)=G_\pi (\Phi,R)$ $(E(R)=E_{\pi}(\Phi, R))$ is an (elementary) Chevalley group of rank $> 1$, $R$ is a local ring (with $\frac{1}{2}$ for the root systems ${\mathbf A}_2, {\mathbf B}_l, {\mathbf C}_l, {\mathbf F}_4, {\mathbf G}_2$ and with $\frac{1}{3}$ for ${\mathbf G}_{2})$, then the group $G(R)$ (or $(E(R)$) is regularly bi-interpretable with the ring~$R$.As a consequence of this theorem, we show that the class of all Chevalley groups over local rings (with the listed restrictions) is elementary definable, i.\,e., if for an arbitrary group~$H$ we have $H\equiv G_\pi(\Phi, R)$, than there exists a ring $R'\equiv R$ such that $H\cong G_\pi(\Phi,R')$.

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Section: Regular Bi-interpretabilitymentioning

confidence: 99%