Decomposition of matrices from \(\text{SL}_{2}(\mathbb{K}[x, y])\)
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 coun...
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Lugansk National Taras Shevchenko University
2025
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admjournalluguniveduua-article-23622025-01-19T19:44:59Z Decomposition of matrices from \(\text{SL}_{2}(\mathbb{K}[x, y])\) Chapovskyi, Yevhenii Kozachok, Oleksandra Petravchuk, Anatoliy row, special linear group, generators, decomposition 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. Lugansk National Taras Shevchenko University 2025-01-19 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2362 10.12958/adm2362 Algebra and Discrete Mathematics; Vol 38, No 2 (2024): A special issue 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2362/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2362/1275 Copyright (c) 2025 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
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| datestamp_date |
2025-01-19T19:44:59Z |
| collection |
OJS |
| language |
English |
| topic |
row special linear group generators decomposition |
| spellingShingle |
row special linear group generators decomposition Chapovskyi, Yevhenii Kozachok, Oleksandra Petravchuk, Anatoliy Decomposition of matrices from \(\text{SL}_{2}(\mathbb{K}[x, y])\) |
| topic_facet |
row special linear group generators decomposition |
| format |
Article |
| author |
Chapovskyi, Yevhenii Kozachok, Oleksandra Petravchuk, Anatoliy |
| author_facet |
Chapovskyi, Yevhenii Kozachok, Oleksandra Petravchuk, Anatoliy |
| author_sort |
Chapovskyi, Yevhenii |
| title |
Decomposition of matrices from \(\text{SL}_{2}(\mathbb{K}[x, y])\) |
| title_short |
Decomposition of matrices from \(\text{SL}_{2}(\mathbb{K}[x, y])\) |
| title_full |
Decomposition of matrices from \(\text{SL}_{2}(\mathbb{K}[x, y])\) |
| title_fullStr |
Decomposition of matrices from \(\text{SL}_{2}(\mathbb{K}[x, y])\) |
| title_full_unstemmed |
Decomposition of matrices from \(\text{SL}_{2}(\mathbb{K}[x, y])\) |
| title_sort |
decomposition of matrices from \(\text{sl}_{2}(\mathbb{k}[x, y])\) |
| description |
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. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2025 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2362 |
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AT chapovskyiyevhenii decompositionofmatricesfromtextsl2mathbbkxy AT kozachokoleksandra decompositionofmatricesfromtextsl2mathbbkxy AT petravchukanatoliy decompositionofmatricesfromtextsl2mathbbkxy |
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2025-12-02T15:46:05Z |
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2025-12-02T15:46:05Z |
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