Decomposition of matrices from $SL_ 2(K[x, y])$

TL;DR

This paper investigates the structure of matrices in SL_2(K[x,y]) over algebraically closed fields, showing decomposition results for matrices with entries of degree ≤ 2 and providing formulas for homogeneous components.

Contribution

It extends understanding of matrix decompositions in SL_2 over polynomial rings, identifying conditions under which matrices can be expressed as products of elementary matrices.

Findings

Matrices with entries of degree ≤ 2 are decomposable into elementary matrices or known exceptions.

Formulas for homogeneous components of matrix entries are derived.

Results clarify the structure of SL_2(K[x,y]) matrices and their decompositions.

Abstract

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and $\mathbb{K}[x,y]$ the polynomial ring. The group $\text{SL}_{2}\left(\mathbb{K}[x,y]\right)$ 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 $\text{SL}_{r}\left(\mathbb{K}[x_{1},\dots,x_{n}]\right)$ is generated by elementary matrices for $r\ge 3$ and arbitrary $n\geq 2$, the same is true for $n=1$ and arbitrary $r.$ It is proven that any matrix from $\text{SL}_{2}\left(\mathbb{K}[x,y]\right)$ 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.