symmetric functions

A symmetric function of is one whose value is unchanged if are permuted arbitrarily. For example, each of the following is a symmetric function of three variables:

Certain symmetric functions serve as building blocks for all the rest. Let

where the sum is taken over all choices of the indices from .
Then is called the th elementary symmetric function of .

The Symmetric Function Theorem

Every symmetric polynomial function of is a polynomial function of . (And every rational function of is a rational function of .)

E.g. For the elementary symmetric functions are

and the above examples expressed in terms of these are:

Theorem Let be the roots of the polynomial equation

and let be the th elementary symmetric function of the . Then

Explanation. On the LHS of the equation

the coefficient of is ; on the RHS, the coefficient of is times the sum of all products of of the . Thus .

Problem. Find all solutions of the system of equations

Solution. The LHS of each equation is a symmetric function of , , . This suggests that we can use the information given to construct a polynomial equation whose roots are , , . Let

Then and

Finally, yields

from which we find

Thus , , are the roots of the cubic equation

Observe that is one of the roots. Now we can factor to obtain

and so find the complete solution set:

where and are the two complex cube roots of .

source
Lozansky, Rousseau – Winning Solutions (1996)