On 3-matrix factorization of polynomials
Let \(R=K[x_{1},x_{2},\cdots, x_{r}]\) and \(S=K[y_{1},y_{2},\cdots, y_{s}],\) where \(K\) is a field. In this paper, we propose a method showing how to obtain \(3\)-matrix factors for a given polynomial using either the Doolittle or the Crout decomposition techniques that we apply to matrices whose...
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| Datum: | 2026 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Lugansk National Taras Shevchenko University
2026
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2228 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1861680840584462336 |
|---|---|
| author | Fomatati, Yves Baudelaire |
| author_facet | Fomatati, Yves Baudelaire |
| author_sort | Fomatati, Yves Baudelaire |
| baseUrl_str | https://admjournal.luguniv.edu.ua/index.php/adm/oai |
| collection | OJS |
| datestamp_date | 2026-04-05T09:02:49Z |
| description | Let \(R=K[x_{1},x_{2},\cdots, x_{r}]\) and \(S=K[y_{1},y_{2},\cdots, y_{s}],\) where \(K\) is a field. In this paper, we propose a method showing how to obtain \(3\)-matrix factors for a given polynomial using either the Doolittle or the Crout decomposition techniques that we apply to matrices whose entries are not real numbers but polynomials. We also explicitly define the category of \(3\)-matrix factorizations of a polynomial \(f\) whose objects are \(3\)-matrix factorizations of \(f\), that is triplets \((P,Q,T)\) of \(m\times m\) matrices such that \(PQT=fI_{m}\). Moreover, we construct a bifunctorial operation \(\overline{\otimes}_{3}\) which is such that if \(X\) (respectively \(Y\)) is a \(3\)-matrix factorization of \(f\in R\) (respectively \(g\in S\)), then \(X\overline{\otimes}_{3} Y\) is a \(3\)-matrix factorization of \(fg\in K[x_{1},x_{2},\cdots, x_{r},y_{1},y_{2},\cdots, y_{s}]\). We call \(\overline{\otimes}_{3}\) the multiplicative tensor product of \(3\)-matrix factorizations. Finally, we give some properties of the operation \(\overline{\otimes}_{3}\). |
| doi_str_mv | 10.12958/adm2228 |
| first_indexed | 2026-04-06T01:00:02Z |
| format | Article |
| id | admjournalluguniveduua-article-2228 |
| institution | Algebra and Discrete Mathematics |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-04-06T01:00:02Z |
| publishDate | 2026 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-22282026-04-05T09:02:49Z On 3-matrix factorization of polynomials Fomatati, Yves Baudelaire \(3\)-matrix factorizations, polynomial, multiplicative tensor product 15A23, 18A05 Let \(R=K[x_{1},x_{2},\cdots, x_{r}]\) and \(S=K[y_{1},y_{2},\cdots, y_{s}],\) where \(K\) is a field. In this paper, we propose a method showing how to obtain \(3\)-matrix factors for a given polynomial using either the Doolittle or the Crout decomposition techniques that we apply to matrices whose entries are not real numbers but polynomials. We also explicitly define the category of \(3\)-matrix factorizations of a polynomial \(f\) whose objects are \(3\)-matrix factorizations of \(f\), that is triplets \((P,Q,T)\) of \(m\times m\) matrices such that \(PQT=fI_{m}\). Moreover, we construct a bifunctorial operation \(\overline{\otimes}_{3}\) which is such that if \(X\) (respectively \(Y\)) is a \(3\)-matrix factorization of \(f\in R\) (respectively \(g\in S\)), then \(X\overline{\otimes}_{3} Y\) is a \(3\)-matrix factorization of \(fg\in K[x_{1},x_{2},\cdots, x_{r},y_{1},y_{2},\cdots, y_{s}]\). We call \(\overline{\otimes}_{3}\) the multiplicative tensor product of \(3\)-matrix factorizations. Finally, we give some properties of the operation \(\overline{\otimes}_{3}\). Lugansk National Taras Shevchenko University 2026-04-05 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2228 10.12958/adm2228 Algebra and Discrete Mathematics; Vol 41, No 1 (2026) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2228/pdf Copyright (c) 2026 Algebra and Discrete Mathematics |
| spellingShingle | \(3\)-matrix factorizations polynomial multiplicative tensor product 15A23 18A05 Fomatati, Yves Baudelaire On 3-matrix factorization of polynomials |
| title | On 3-matrix factorization of polynomials |
| title_full | On 3-matrix factorization of polynomials |
| title_fullStr | On 3-matrix factorization of polynomials |
| title_full_unstemmed | On 3-matrix factorization of polynomials |
| title_short | On 3-matrix factorization of polynomials |
| title_sort | on 3-matrix factorization of polynomials |
| topic | \(3\)-matrix factorizations polynomial multiplicative tensor product 15A23 18A05 |
| topic_facet | \(3\)-matrix factorizations polynomial multiplicative tensor product 15A23 18A05 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2228 |
| work_keys_str_mv | AT fomatatiyvesbaudelaire on3matrixfactorizationofpolynomials |