We are going to be dealing with polynomials in n variables x1, x2, ..., xn. Such a polynomial is called symmetric if any rearrangement of the variables leaves the polynomial unchanged.

Formally, a polynomial f(x1,...,xn) is symmetric iff f(x1,...,xn)=f(xs(1),...,xs(n)) for any permutation s in the nth Symmetric group Sn

For example,

  • If n=1, so there is just one variable, then every polynomial is symmetric.
  • If n=2, then
    • x1 + x2 is symmetric.
    • x1x2 is symmetric.
    • x1 + (x2)2 is not. (Because when we swap over x1 and x2 for this polynomial we get x2 + (x1)2.)
  • If n=3, then
    • x1 + x2 is not symmetric because if we consider the permutation that sends 1 -> 2 -> 3 and apply it to our polynomial we get x2 + x3.
    • x1 + x2 + x3 is symmetric though and so is (x1)2 + (x2)2 + (x3)2.

The elementary symmetric polynomials are defined as follows:

  • First we take the sum of all the variables
    s1=Sum(1<= i <=n) xi. Thus, s1=x1+x2 + ... + xn.
  • Next we take the sum of all products of pairs of distinct variables
    s2=Sum(1<= i < j <=n) xixj.
  • Then we take the sum of all products of three distinct variables
    s3=Sum(1<= i < j < k <=n) xixjxk.
  • ...
  • Finally we take the product of all of the variables sn=x1...xn

It is not too hard to see that s1, ..., sn are symmetric polynomials.

In the case n=2 we get

  • s1=x1 + x2.
  • s2=x1x2.

For n=3 we get

  • s1=x1 + x2 + x3.
  • s2=x1x2 + x1x3 + x2x3.
  • s3=x1x2x3.

It turns out that any symmetric polynomial can be constructed out of these elementary ones.

Theorem Let f(x1,...,xn) be a symmetric polynomial over a field then there exists a polynomial g(x1,...,xn) such that f(x1,...,xn)=g(s1,...,sn).

Here is a proof of the fundamental theorem on symmetric polynomials.

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