POLYNOMIAL MULTIVARIATE DECOMPOSITION

Jaime Gutierrez and Rosario Rubio San Miguel

Dpto. Matemáticas, Estadística y Computación

Universidad de Cantabria, Santander 39071, Spain

$\{ sarito, jaime \}$ @matesco.unican.es

Abstract

During the last years several results has been obtained on the univariate polynomial decomposition area. However, multivariate decomposition problem has not been studied so much.

Generalizing the concept of univariate decomposition by intermediate field theory, we could say that a multivariate polynomial $f(X_1,\dots,X_n)$ is decomposable if and only if there exists an intermediate field ${\rm I\kern -2.2pt F\hskip 1pt}$ such that $I\!\!K(f) \subset {\rm I\kern -2.2pt F\hskip 1pt}\subset I\!\!K(X_1,\dots,X_n)$. Such field has the following form: $I\!\!K(h_1,\dots,h_m)$, so f can be decomposed as $f = g( h_1,\dots,h_m ).$ Therefore this problem is divided into two parts: the first one is the computation of $h_1,\dots,h_m$and afterwards, the computation of g given $(h_1,\dots,h_m)$.

A complete solution to this problem seems to be more difficult than the decomposition for univariate rational functions. But even ``non-trivial" partial solutions would be an aid for algebraic simplification and evaluation questions. We study three different decomposition problems that appears in the literature. These problems comes out, when we set some restrictions over the field ${\rm I\kern -2.2pt F\hskip 1pt}$:

** ${\rm I\kern -2.2pt F\hskip 1pt}$ is a 1-transcendental degree field. Therefore, there exist g(Y) and $h(X_1,\dots,X_n)$ such that:

$$f(X_1,\dots,X_n) = g(h(X_1,\dots,X_n)).$$

** ${\rm I\kern -2.2pt F\hskip 1pt}$ is generated by non-constant polynomials of $I\!\!K(X_i)$, for all $i=1,\dots,n$. Therefore, there exist $g(Y_1,\dots,Y_n)$and univariate non-constant polynomials h_i(X_i), such that:

$$f(X_1,\dots,X_n) = g(h_1(X_1),\dots,h_n(X_n)).$$

** There exist $i \in \{1,\dots,n \}$ such that $I\!\!K(X_1,\dots,X_{i-1},f,X_{i+1},\dots,X_n) \subset {\rm I\kern -2.2pt F\hskip 1pt}\subset I\!\!K(\underline{X}).$ In this problem, there exist $g(Y_1,\dots,Y_n)$ and $h(X_1,\dots,X_n)$such that

$$f=g(X_1,\dots ,X_{i-1},h(X_1,\dots ,X_n),X_{i+1},\dots ,X_n).$$