**Jaime Gutierrez and Rosario Rubio San Miguel
Universidad de Cantabria
Departamento de Matematicas, Estadistica y Computacion
Facultad de Ciencias, Avda. Los Castros s/n
Santander, Cantabrria, 39005
Spain
E-mail:jaime@matesco.unican.es**

**Abstract**

The functional decompositon problem for univariate
rational functions can be stated as follows (see Zippel (1991)):
given an univariate rational function *f*(*x*) with coefficients in
an arbitrary field ,
,
determine whether there exist two rational functions
with
degree larger than one, such that
*f*(*x*)=*g*(*h*(*x*)), and, in the affirmative case,
compute them. This problem has several applications, for instance,
faithful re-parameterizing unfaithfully
parameterized curves, providing a birationality test for subfields of ,
computing intermediate fields in a simple purely transcendental
field extension ,
etc.

The method in Alonso, Gutierrez and Recio (1995) for
decomposing a univariate rational function used the concept of near-separated
polynomial: to each univariate rational function
*f*(*x*)=*f*_{n}(*x*)/*f*_{d}(*x*),
we associate the near-separated polynomial
*f*_{n}(*x*)*f*_{d}(*y*)-*f*_{n}(*y*)*f*_{d}(*x*), The key of the algorithm is based
on the following:

*``Given two
rational functions
*
*f*(*x*), *h*(*x*)*, there exists a rational function
**g*(*x*)* such that *
* if and only if the near-separated polynomial
associated to **h** divides the one associated to **f**"*.

The generalization of the decomposition problem to multivariate rational functions seems more involved. Similarly, we translate the above idea to multivariate functional decomposition. Now, to each list of multivariate rational functions , we consider the near-separated ideal, Ideal , of the polynomial ring . Using this new concept and the Gröbner basis theory, we get the following theorem. In the polynomial ring we have:

*``If the
near-separated ideal associated to *
* is a subset of the
one associated to *
*, then
*
*."*

The converse holds if we consider the near-separated ideals
over
,
the ring
of fractions of
with denominator *S*, the multiplicative
closed
set generated by all the polynomials in the variables
and
.

These results shed a new light on multivariate decomposition problem with application to compute intermediate fields in a purely transcendental field extension , which is a classical topic in Algebra (see Sweedler (1993)).

IMACS ACA'98 Electronic Proceedings