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On 3-matrix factorizations of polynomials

Yves Baudelaire Fomatati

Abstract

Let $R=K[x_{1},x_{2},\cdots, x_{m}]$ and $S=$ $K[y_{1},y_{2},\cdots, y_{m}]$ where $K$ is a field. %commutative ring with unity. 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 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 $\bar{\otimes}_{3}$ which is such that if $X$ (respectively $Y$) is a $3-$matrix factorization of $f\in R$ (respectively $g\in S$), then $X\bar{\otimes}_{3} Y$ is a $3-$matrix factorization of $fg\in K[x_{1},x_{2},\cdots, x_{m},y_{1},y_{2},\cdots, y_{m}]$. We call $\bar{\otimes}_{3}$ the multiplicative tensor product of $3-$matrix factorizations. Finally, we give some properties of the operation $\bar{\otimes}_{3}$.

On 3-matrix factorizations of polynomials

Abstract

Let and where is a field. %commutative ring with unity. In this paper, we propose a method showing how to obtain -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 define the category of -matrix factorizations of a polynomial whose objects are -matrix factorizations of , that is triplets of matrices such that . Moreover, we construct a bifunctorial operation which is such that if (respectively ) is a matrix factorization of (respectively ), then is a matrix factorization of . We call the multiplicative tensor product of matrix factorizations. Finally, we give some properties of the operation .
Paper Structure (12 sections, 5 theorems, 19 equations)

This paper contains 12 sections, 5 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Doolittle and Crout matrix decomposition methods
  4. $n$-matrix factorization of polynomials for $n=2$;$3$
  5. 2-matrix factorization of polynomials
  6. 3-matrix factorizations and examples
  7. The category of 3-matrix factorizations of $f\in R$
  8. The multiplicative tensor product of 3-matrix factorizations
  9. Definition and examples
  10. Funtoriality of $\overline{\otimes}_{3}$
  11. Properties of $\overline{\otimes}_{3}$: Associativity, commutativity and distributivity
  12. FURTHER PROBLEMS

Key Result

Lemma 5.1

$\Phi_{f}\overline{\otimes}_{3} \Phi_{g}$: $X_{f}\overline{\otimes}_{3}X_{g} =(\phi,\psi,\theta)\overline{\otimes}_{3} (\sigma,\rho,\zeta)\rightarrow X_{f}'\overline{\otimes}_{3}X_{g}' =(\phi',\psi',\theta')\overline{\otimes}_{3} (\sigma',\rho',\zeta')$ is a morphism in $MF(K[x,y],fg)_{3}$.

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 3.1
  • Example 3.1
  • Definition 3.2
  • Example 3.2
  • Remark 3.1
  • Example 3.3
  • Definition 5.1
  • Example 5.1
  • Definition 5.2
  • ...and 10 more