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Mathematics & Statistics

Classical Analysis
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Tuesday, August 3, 2010



A solution to the Gromov-Vaserstein Problem
Frank Kutzschebauch
University of Berne
PHY 108


Any matrix in \(\operatorname{Sl}_n\,(\mathbb{C})\) can (due to the Gauss elimination process) be written as a product of elementary matrices. If instead of the complex numbers (a field) the entries in the matrix are elements of a ring, this becomes a delicate question. In particular the rings of maps from a space \(X\to\mathbb{C}\) are interesting cases. A deep result of Suslin gives an affirmative answer for the polynomial ring in \(m\) variables in case the size of the matrix (\(n\)) is greater than 2. In the topological category the problem was solved by Thurston and Vaserstein. For holomorphic functions on \(\mathbb{C}^m\) the problem was posed by Gromov in the 1980's. We report on a complete solution to Gromov's problem. A main tool is the Oka-Grauert-Gromov-\(h\)-principle in Complex Analysis. This is joint work with Björn Ivarsson.