A problem with rational functions

[Nicola Sottocornola]

Hi,

I have six rational functions f1,...,f6 in 4 coordinates:

f_i \in Q(x,y,z,w) for i=1..6.

I want to know if one of them, say f1 as an example, belongs to the
field Q(f2,...,f6) generated by the other 5 functions.

Can Maple do this?

Thanks, Nicola

[Dave Rusin]

In article <2Cq%g.186$Pk4...@tornado.fastwebnet.it>,
Nicola Sottocornola <Nicola.So...@fastwebnet.it> wrote:

Sure, at least in principle. Introduce 6 new variables g_1, ..., g_6,
and then issue the command
el:=eliminate({seq( f||i - g||i, i=1..6)}, {x,y,z,w} );
Then el[1] will give you equations expressing each of x,y,z,w in
terms of g1 through g6; el[2] will give you equations expressing
the algebraic relations among those six g's. You need only see
whether it is possible to solve for g1 in terms of the other 5.

Generically you would expect el[2] to consist of two polynomials.

I tried a simpler example: 4 rational functions in 2 variables. This
already seems to be too hard for Maple. I used
f1 := (3+2*x-4*y+7*x*y)/(1-x-y) ;
f2 := (13-2*x+5*y^2-8*x^3*y)/(x+x*y+y^3) ;
f3 := (2*x^2-6*x*y^2-y^4)/(11-y^2+x*y^4) ;
f4 := (1+x+x^2+x^3)/(1-y-y^2) ;

Even in this simpler case I was unable to obtain a solution within the few
minutes I was willing to wait. (On the other hand if I change this to a
1-variable problem by setting, say, y:=23, then it is true that each of
f2, f3, and f4 lies in the same subfield as f1, namely
f2 = 6*(326989*f1^3-3744004*f1^2+51527473*f1+1896113658)/(11639*f1+1985357)/
(f1+163)^2 ,
f3 = 1/3*(41809*f1^2+16027304*f1+1496224981)/(f1+163)/(410468*f1-1654761) ,
f4 = 105/551*(f1-12)*(97*f1^2-718*f1+6898)/(f1+163)^3

It might help a lot if you had some other understanding of what this
subfield of Q(x,y,z,w) is -- for example, if it is the subfield invariant
under some group action or something. (I can try to work on it if you will
send me the rational functions and some background information.)

dave