POLYNOMIAL MULTIVARIATE DECOMPOSITION

Jaime Gutierrez and Rosario Rubio San Miguel

Dpto. Matemáticas, Estadística y
Computación

Universidad de Cantabria, Santander 39071, Spain

@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
is *decomposable* if and
only if there exists
an intermediate field
such that
.
Such field has the following form:
,
so *f* can be decomposed as
Therefore this problem
is
divided into two parts: the first one is the computation of
and afterwards, the computation of *g* given
.

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 :

**
is a 1-transcendental degree field. Therefore, there exist
*g*(*Y*) and
such that:

**
is generated by non-constant polynomials of
,
for all
.
Therefore, there exist
and
univariate non-constant polynomials *h*_{i}(*X*_{i}), such that:

** There exist such that In this problem, there exist and such that

IMACS ACA'98 Electronic Proceedings