I am looking for a version of Suslin's Stability Theorem for Chevalley groups. 
The version of the theorem for $G=SL\_n({\mathbb Z}[x\_1, \dots , x\_m])$ states that the if $n\ge m+2$, the elementary matrices generate $G$. I think by embedding many copies of $SL\_n$, one can prove a similar statement, but probably with a bad bound, for all Chevalley groups over ${\mathbb Z}[x\_1, \dots , x\_m]$. I was wondering if a sharp bound has already been worked out in the literature. 



Suslin's Stability Theorem for Chevalley Groups
I am somewhat puzzled by your version of "Suslin stability theorem". What you are referring to is a combination of usual stabilization for $SK\_1$ and an estimate for the stable rank of integer polynomials. Suslin's theorem states something much stronger, in particular Corollary 6.6 in his paper [On the structure of the special linear group over polynomial rings](http://mi.mathnet.ru/eng/izv/v41/i2/p235) is as follows (note that it doesn't depend on the number of variables):



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> Let $A$ be a regular ring such that $SK\_1(A)=0$ (for example, the ring of integers in an algebraic number field). Then $SL\_r(A[x\_1,\ldots,x\_n])$ is generated by elementary matrices for $r\geqslant\max(3,\dim A+2)$.
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Corollary 7.10 extends this result to rings of the form $A[x\_1^\pm,\ldots,x\_k^\pm,x\_{k+1},\ldots,x\_n]$ for $A$ regular.


For other Chevalley groups the situation is complicated. One has the stability theorems for $K\_1(\Phi)$ in terms of stable rank (or its ramifications such as absolute stable rank or $\Lambda$-stable rank), but they give pretty bad bounds for polynomial rings.


There is, however, the following version of Suslin's theorem for symplectic group in a paper [On symplectic groups over polynomial rings](https://doi.org/10.1007/BF02571323) by F. Grunewald, J. Mennicke and L. Vaserstein:



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> Let $A$ be a locally principal ring. For an integer $m\geqslant0$ put $R=A[x\_1,\ldots,x\_m]$. Then $Sp\_{2n}(R)=Sp\_{2n}(A)\cdot Ep\_{en}(R)$ for any $n\geqslant2$.
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By locally principal ring they mean a commutative ring such that its localization at any maximal ideal is a principal ideal ring.


For euclidean ring $A$ this gives $K\_1(\mathsf{C}\_\ell,R)=0$.


As a byproduct they also prove a stronger version of Suslin's theorem for $SL$ and a locally principal ring.


They also have a version for Laurent polynomial rings and claim that by using stability theorems for $K\_1$ as in M. Stein's papers one can prove the same results for classical simple algebraic groups of relative rank $\geqslant2$, but the latter has never been written in full details.


**UPDATE 14.02.2019**
The proof for all simply-connected Chevalley groups is given by A. Stavrova in [Chevalley groups of polynomial rings over Dedekind domains](https://arxiv.org/abs/1812.04326). Namely, she proves the following theorem:



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> Let $R$ be a locally principal ring, and let $G$ be a Chevalley—Demazure group scheme of isotropic rank $\geq2$. Then $G(R[x\_1,\ldots,x\_n])=G(R)E(R[x\_1,\ldots,x\_n])$ for any $n\geq1$.
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If $R$ is a Dedekind ring of arithmetic type (for example, $R=\mathbb{Z}$), it follows that $G(R[x\_1,\ldots,x\_n])=E(R[x\_1,\ldots,x\_n])$.


