SL_n(F[x]) is not boundedly generated by elementary matrices: explicit proof

Abstract

Using methods of higher algebraic K-theory, van der Kallen proved that SL_n(F[x]) does not have bounded word length with respect to elementary matrices if the field F has infinite transcendence degree over its prime subfield. A short explicit proof of the result is exhibited by constructing a sequence of matrices with infinetely growing word length. This construction is also used to show that SL_n(Z[x]) does not have bounded word length with respect to elementary matrices of "bounded degree".