ELA, Volume 11, pp. 162-167, June 2004, abstract.

SL_n(F[x]) Is Not Boundedly Generated by Elementary Matrices

Igor V. Erovenko

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 this result is exhibited by constructing a sequence of
matrices with infinitely 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".