An elementary transformation of a special unimodular vector to its top coefficient vector

Abstract:

Let $R$ be a commutative ring, ${\mathbf {v}}(X)$ a unimodular $n$-vector $(n \geqslant 3)$ over $R[X]$. Suppose the leading coefficients in ${\mathbf {v}}(X)$ form a unimodular vector $L({\mathbf {v}})$ over $R$. Then some element in ${E_n}(R[X])$ will transform ${\mathbf {v}}(X)$ to $L({\mathbf {v}})$.