摘要：It is standard material in a Linear Algebra course that the group $\mbox{SL}_m(\mathbb{C})$ is generated by elementary matrices $E+ \alpha e_{ij} \ i\ne j$, i.e., matrices with 1's on the diagonal and all entries outside the diagonal are zero, except one entry. 

The same question for matrices in $\mbox{SL}_m(R)$ where $R$ is a commutative ring instead of the field $\mathbb{C}$ is much more delicate. Especially interesting is the case that $R$ is the ring of complex valued functions (continuous, smooth, algebraic or holomorphic) from a space $X$.

For $m\ge 3$ (and any $n$) it is a deep result of \textsc{Suslin} that any matrix in $\mbox{SL}_m(\mathbb{C}[\mathbb{C}^n])$ decomposes as a finite product of unitriangular (and equivalently elementary) matrices. 

In the case of continuous complex valued functions on a topological space $X$ the problem was studied and  solved  by \textsc{Thurston} and \textsc{Vaserstein}. For rings of holomorphic functions on Stein spaces, in particular on $\mathbb{C}^n$, this problem was explicitly posed as the {\bf Vaserstein problem} by \textsc{Gromov} in the 1980's.

In the talk we explain a complete solution to \textsc{Gromov's} {\sl Vaserstein Problem} from a joint work with B. Ivarsson. We indicate the method of the proof: A very advanced version of the Oka-principle proposed by \textsc{Gromov} and proved in recent years by \textsc{Forstneri\v{c}}: An elliptic stratified submersion over a Stein space admits a holomorphic section iff it admits a continuous section.