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

2024/12/04 by Y. Chapovskyi, Chapovskyi, Y., Oleksandra Kozachok +3
Mathematics · Computer Science · #Advanced Topics in Algebra #Matrix Theory and Algorithms #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2412.03688

Abstract

Let \mathbbK be an algebraically closed field of characteristic zero and \mathbbK[x,y] the polynomial ring. The group SL₂(\mathbbK[x,y]) of all matrices with determinant equal to 1 over \mathbbK[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 SL_r(\mathbbK[x₁,…,x_n]) is generated by elementary matrices for r≥ 3 and arbitrary n≥ 2, the same is true for n=1 and arbitrary r. It is proven that any matrix from SL₂(\mathbbK[x,y]) with at least one entry of degree ≤ 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 \beginpmatrixf · amp; g
-Q · amp; P\endpmatrix\inSL₂(\mathbbK[x,y]), 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.

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