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.


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.





x + y = u

xy = v, we have   



By x in R,   

    u² - 4v ≧ 0