Return to Question

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 field.

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 $\mathrm{sr}(\mathbf Z[X]) \not \leq 2$?

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 $\mathrm{sr}(\mathbf Z[X]) \not \leq 2$?

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:

$$1 \in \langle a, a_1, \dots, a_{n+1}\rangle \implies \exists \ x_1, \dots, x_n \in A, 1 \in \langle a_1+x_1a,\ \dots, a_n + x_n a\rangle.$$ (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 field.

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 $\mathrm{sr}(\mathbf Z[X]) \not \leq 2$?

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 field.

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 $\mathrm{sr}(\mathbf Z[X]) \not \leq 2$?

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:

$$1 \in \langle a, a_1, \dots, a_{n+1}\rangle \implies \exists \ x_1, \dots, x_n \in A, 1 \in \langle a_1+x_1a,\ \dots, a_n + x_n a\rangle.$$

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 field.

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 $\mathrm{sr}(\mathbf Z[X]) \not \leq 2$?

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:

$$1 \in \langle a, a_1, \dots, a_{n+1}\rangle \implies \exists \ x_1, \dots, x_n \in A, 1 \in \langle a_1+x_1a,\ \dots, a_n + x_n a\rangle.$$ (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 field.

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 $\mathrm{sr}(\mathbf Z[X]) \not \leq 2$?