Symmetric Polynomials

Date: 07/31/97 at 05:43:03
From: Jillian Stewart
Subject: Symmetric polynomials

Hello!

I have a question about polynomials and I hope you can help me (no one 
else could).

How can it be proved that any symmetric polynomial can be expressed as 
an elementary symmetric polynomial?  (I don't even know what symmetric 
polynomials are.)

Thank you!

Date: 07/31/97 at 11:36:48
From: Doctor Wilkinson
Subject: Re: Symmetric polynomials

A symmetric polynomial is one that doesn't change when you permute its
variables. For example, x^2 + y^2 is a symmetric polynomial in x and 
y, because if you exchange x and y, the polynomial doesn't change; but
x^2 + 2y^2 isn't symmetric, because if you exchange x and y you get
2x^2 + y^2, which is different.

The elementary symmetric polynomials in x1, x2, x3, x4, ..., xn are

 s1 = x1 + x2 + x3 + ... + xn
 s2 = x1x2 + x1x3 +... +x2x3 + x2x4 +... + ... +... +xn-1xn
 ...
 sn = x1x2x3...xn

That is, the kth symmetric polynomial is formed by taking all the 
possible products of k variables and adding them together.

The theorem you cite is non-trivial but it is not really hard either.  
It can be proved by induction on the number of variables and you can 
find a proof in a lot of advanced algebra books.  An example of what 
it is saying is that

 x^2 + y^2 = (x + y)^2 - 2xy = (s1(x, y))^2 - 2s2(x, y)

If you try some other examples, you can sort of figure out how a 
general proof has to go, but you have to be pretty good at induction 
proofs to get it all to fit together.

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/