Decomposition of matrices from \(\text{SL}_{2}(\mathbb{K}[x, y])\)

Abstract

Let \(\mathbb{K}\) be an algebraically closed field of characteristic zero and \(\mathbb{K}[x, y]\) the polynomial ring. The group \(\text{SL}_{2}(\mathbb{K}[x, y])\) of all matrices with determinant equal to \(1\) over \(\mathbb{K}[x, y]\) can not be generated by elementary matrices. The known counterexample was pointed out by P. M. Cohn. Conversely, A. A. Suslin proved that the group \(\text{SL}_{r}(\mathbb{K}[x_1, . . . , x_n])\) is generated by elementary matrices for \(r\geq 3\) and arbitrary \(n\geq 2\), the same is true for \(n = 1\) and arbitrary \(r\). It is proven that any matrix from \(\text{SL}_{2}(\mathbb{K}[x, y])\) with at least one entry of degree \(\le 2\) is either a product of elementary matrices or a product of elementary matrices and of a matrix similar to the one pointed out by P. Cohn. For any matrix \(\begin{pmatrix}\begin{array}{cc} f & g\\ -Q & P \end{array}\end{pmatrix}\in\text{SL}_{2}\left(\mathbb{K}[x,y]\right)\), we obtain formulas for the homogeneous components \(P_i , Q_i\) for the unimodular row \((-Q, P)\) as combinations of homogeneous components of the polynomials \(f, g,\) respectively, with the same coefficients.