From my understanding, when computing with huge rational functions, we shouldn't use symbolic expressions. However, I don't see how to find roots and residues without symbolic variables. Here is a small scale example of the issue. I have a rational function that looks like below:

$f(u_1,x_1,u_2,x_2,u_3,x_3) = \frac{1}{{\left(u_{1} u_{2} u_{3} - x_{1} x_{2} x_{3}\right)} {\left(u_{1} u_{2} u_{3} - 1\right)} {\left(u_{1} u_{2} - x_{1} x_{2}\right)} {\left(u_{1} u_{2} - 1\right)} {\left(u_{1} - x_{1}\right)} {\left(u_{1} - 1\right)}}$

First I want to solve for $u_1$ in the denominator and find those roots (poles) that have $x_1$ as below.

Then I will loop through the roots and compute the residue of $f$ w.r.t. $u_1$ at those poles.

```
u1,u2,u3,x1,x2,x3 = var('u1,u2,u3,x1,x2,x3') #symbolic variables
f=1/((u1*u2*u3 - x1*x2*x3)*(u1*u2*u3 - 1)*(u1*u2 - x1*x2)*(u1*u2 - 1)*(u1 - x1)*(u1 - 1))
fden=f.denominator() #denominator of f
list1=fden.roots(u1) #poles of u1
[root for (root, multiplicity) in list1] #list of roots
```

The output is

```
[x1*x2*x3/(u2*u3), x1*x2/u2, x1, 1/(u2*u3), 1/u2, 1]
```

Then we choose those roots that have $x1$

```
poles1 = [x1*x2*x3/(u2*u3), x1*x2/u2, x1] #choose those that have x1
```

Finally, we find the residue of $f$ w.r.t $u1$ of the rational function at the poles containing $x1.$

```
ans1=0
for ff in poles1:
tmp=f.residue(u1==ff)
ans1+=tmp #ans1 is the residue of f w.r.t u1 at all the poles containing x1
ans1
```

The output is then

```
1/((u2*u3*x1 - x1*x2*x3)*(u2*u3*x1 - 1)*(u2*x1 - x1*x2)*(u2*x1 - 1)*(x1 - 1)) - 1/((u3*x1*x2 - x1*x2*x3)*(u3*x1*x2 - 1)* (x1*x2 - 1)*u2*(x1 - x1*x2/u2)*(x1*x2/u2 - 1)) + 1/((x1*x2*x3 - 1)*(x1*x2 - x1*x2*x3/u3)*(x1*x2*x3/u3 - 1)*u2*u3*(x1 x1*x2*x3/(u2*u3))*(x1*x2*x3/(u2*u3) - 1))
```

Then I replace $f$ with $ans1$ continue to do the same process w.r.t $u2$ and poles containing $x2$ and finally w.r.t $u3$ and poles containing $x3.$ However, this consumes about 800GB of memory on an HPC when I feed it a larger rational function.

Is there a way to find

roots of multivariable polynomials with respect to one variable ?

residue of a rational function avoiding symbolic variables?

Both .roots() and .residue() are not defined for rational functions that are not defined in terms of symbolic variables.