Symmetry expression

For x, y in C,

 

Consider simultaneous equations of x and y

x + y = u

xy = v.

 

Let the solutions be x = α and y = β.

Thus

α+β = u

αβ = v.

 

A quadratic equation whose solutions are α and β is

t² -ut+v =0.

 

By α, β in C, u and v are also in C.

 

Here, considering α, β in R, the solutions of  t² -ut+v =0 must be real solutions.

Hence u²-4v≧0 holds.