What are Symmetric Functions for Polynomial Roots?

[Jhenrique]

My question is hard of answer and the partial answer is in the wikipedia, but maybe someone known some article that already approach this topic and the answer is explicited. So, my question is:

given:
$A = x_1 + x_2$
$B = x_1 x_2$

reverse the relanship:
$x_1 = \frac{A + \sqrt[2]{A^2-4B}}{2}$
$x_2 = \frac{A - \sqrt[2]{A^2-4B}}{2}$

So, given
$A = x_1 + x_2 + x_3$
$B = x_2 x_3 + x_3 x_1 + x_1 x_2$
$C = x_1 x_2 x_3$

and:
$A = x_1 + x_2 + x_3 + x_4$
$B = x_1 x_2 + x_1 x_3 + x_1 x_4 + x_2 x_3 + x_2 x_4 + x_3 x_4$
$C = x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4$
$D = x_1 x_2 x_3 x_4$

thus which would be the inverse relationship for those two systems above?

 

[The_Duck]

So, given
$A = x_1 + x_2 + x_3$
$B = x_2 x_3 + x_3 x_1 + x_1 x_2$
$C = x_1 x_2 x_3$

Write

$(x - x_1)(x - x_2)(x - x_3) = x^3 - Ax^2 + Bx - C = 0$

Then use the general solution to the cubic equation to solve for the roots, which are $x_1, x_2, x_3$.

You can similarly turn your four-variable case into a quartic equation.

 

[bhillyard]

I think what you are looking for is
'symmetric functions' of the roots of a polynomial equation.
If the roots of a cubic are x₁, x₂, x₃ then the equation is:

(x - x₁)(x - x₂)(x - x₃) = 0

Multiply out the brackets, gather together the terms in x³, x², x and the constant.

You will find your expressions x₁ + x₂ + x₃, x₁x₂+x₂x₃+x₃x₁, x₁x₂x₃

appearing as the coefficients of the powers of x.

This pattern continues with roots of quadratic, Quartic etc.