# Efficient reduction to elementary symmetric polynomials

When using the resolvent method for solving polynomial equations, there's a step to express the coefficients of a polynomial (so-called resolvent) in elementary symmetric polynomials. For instance, for the general quintic one gets a resolvent of degree six, and one of its simpler coefficients is:

```
15*e[1, 1, 1, 1, 1, 1, 1, 1] - 160*e[2, 1, 1, 1, 1, 1, 1] + 528*e[2, 2, 1, 1, 1, 1] - 448*e[2, 2, 2, 1, 1] - 160*e[2, 2, 2, 2] + 336*e[3, 1, 1, 1, 1, 1] - 2208*e[3, 2, 1, 1, 1] + 2752*e[3, 2, 2, 1] + 2368*e[3, 3, 1, 1] - 3328*e[3, 3, 2] - 96*e[4, 1, 1, 1, 1] + 512*e[4, 2, 1, 1] + 2304*e[4, 2, 2] - 8960*e[4, 3, 1] + 17920*e[4, 4] - 1136*e[5, 1, 1, 1] + 5456*e[5, 2, 1] - 14144*e[5, 3] + 2736*e[6, 1, 1] - 6464*e[6, 2] + 1248*e[7, 1] + 1664*e[8]
```

The problem is that the reduction of the more complicated coefficients takes quite long even for this small degree polynomial. There are, however, two special properties of my polynomial that I think could be utilized here:

- My input polynomial is in the depressed form, meaning that $e_1$ is 0. Therefore I can ignore all terms where $e_1$ appears.
- I know that I'm working with symmetric expressions of 5 roots, therefore I can also ignore all terms with $e_i$ for $i > 5$.

Is there any way how to utilize those properties to help Sage simplify the computation? I'm using the `from_polynomial`

function from the `SymmetricFunctions`

class.