On 3-matrix factorization of polynomials

• Date: 2026
• Main Author: Fomatati, Yves Baudelaire
• Format: Article
• Language: English
• Published: Lugansk National Taras Shevchenko University 2026
• Subjects: \(3\)-matrix factorizations; polynomial; multiplicative tensor product; 15A23; 18A05
• Online Access: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2228
• Journal Title: Algebra and Discrete Mathematics

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 \(\overline{\otimes}_{3}\).