Let $n$ be a positive integer and $A$ be a commutative ring. The ring $A$ is said to be of Bass stable range $\mathrm{sr}(A)\leq n$ if for $a, a\_1, \dots, a\_n \in A$ one has the following implication:


$$1 \in \langle a, a\_1, \dots, a\_{n}\rangle \implies \exists \ x\_1, \dots, x\_n \in A, 1 \in \langle a\_1+x\_1a,\ \dots, a\_n + x\_n a\rangle.$$
(Above, $\langle \cdot \rangle$ denotes the ideal generated by the elements inside).


Of course, one says that $A$ is of stable range $n$ if $\mathrm{sr}(A) \leq n$ and $\mathrm{sr}(A) \not \leq n-1$.


Bass proved that if $A$ is noetherian of Krull dimension $d$ then $\mathrm{sr}(A)\leq d+1$.


Examples are known; for example Vaserstein proved that $\mathrm{sr}(k[x\_1,\dots,x\_n]) = n+1$ when $k$ is a subfield of the real numbers. 


My question is : is the stable range of the ring of integer polynomials $\mathbf Z[X]$ known?


What I wrote before shows that $\mathrm{sr}(\mathbf Z[X]) \leq 3$ and it seems very likely that $\mathrm{sr}( \mathbf Z[X])=3$.


A refinement of my previous question is : could you provide an explicit unimodular triplet of polynomials $(P\_1,P\_2,P\_3) \in \mathbf Z[X]$ showing that $\mathrm{sr}(\mathbf Z[X]) \not \leq 2$?



Bass' stable range of $\mathbf Z[X]$
Sorry for putting this in a separate answer, but I think it will be cleaner this way.



> 
> The stable range of $\mathbb{Z}[x]$ is equal to 3.
> 
> 
> 


I believe I now understand Vaserstein's intended argument that it's $\ge 3$:


(1) There are rings of the form $A={\mathbb Z}[x]/(h)$ such that $SK\_1(A)\neq 0$. One way to get such a ring is to start with the ring of integers in a quadratic field, let $I$ be a "sufficiently small" ideal (I confess to not being exactly sure what this means) and look at the subring generated by $1$ and $I$. For some definition of "sufficiently small", Bass has shown that this gives us $SK\_1(A)\neq 0$. 


(2) Take a non-zero element of $SK\_1(A)$ and represent it by a Mennicke symbol $[\overline{f},\overline{g}]$. 


(3) Then $(f,g,h)$ is a unimodular row over $\mathbb{Z}[x]$. 


(4) Clearly, the Mennicke symbol $[\overline{f},\overline{g}]$ does not lift to $K\_1({\mathbb Z}[x])$.


(5) It follows from Lemma 17.1 of the paper referenced by Jeremy Rickard that the row $(f,g,h)$ is not reducible. 


I'm still just a tad unclear on why it is, in point (1), that we can take the kernel of ${\mathbb Z}[x]\rightarrow A$ to be principal. Is this obvious? I'll add a comment if I nail this down.


