Symmetric Polynomial & Elementary Symmetric Functions

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What is a Symmetric Polynomial?

A polynomial is a symmetric polynomial if its variables are unchanged under any permutation (i.e. swap). In other words, if you switch out two of the variables, you end up with the same polynomial.

Examples

The polynomial x + y + z is symmetric because if you switch any of the variables, it remains the same. In other words,

x + y + z = y + z + x = z + x + y

The polynomial x1x3 + 3x1x2x3 is a symmetric polynomial, because if you swap the variables, then it’s still the same polynomial. For example, the following image shows that swapping x1 and x3 results in the same polynomial:

In other words, x1x3 + 3x1x2x3 is the same polynomial as x3x1 + 3x3x2x1

On the other hand, x1x2 + x2x3 is not symmetric. If you swap two of the variables (say, x2 and x3, you get a completely different expression.

Elementary Symmetric Polynomial

Elementary symmetric polynomials (sometimes called elementary symmetric functions) are the building blocks of all symmetric polynomials. For the variables x1, x2, x3,… xn, they are defined mathematically as follows:

• S1 = x1 + x2 + x3 + … + xn
• S2 = x1 x2 + x1 x3 + x1 x4 + … + x(n – 1) x(n – 2)
• S3 = x1 x2 x3 + x1 x2 x4 + … + x(n – 2)x(n – 1)xn
• Sn = x1 x2 x3 … xn

As an example, the elementary symmetric polynomials for the variables x1, x2 and x3 are:

1. S1 = x1 + x2 + x3
2. S2 = x1 x2 + x1 x3 + x2 x3
3. S3 = x1 x2 x3

Graphs of a Symmetric Polynomial

They are called “symmetric” not because their graph shows symmetry, but because they remain the same if you permute their roots. As far as graphing, the graph of any symmetric polynomial would look exactly the same no matter which variables you switch around. If the graph changes, then the expression is not symmetric.

Why are Symmetric Polynomials Important?

Symmetric polynomials are particularly important in number theory because two types—the elementary symmetric and power sum symmetric polynomials can completely represent any set of points in the set of all complex numbers.

References

Egge, E. (2018). Combinatorics of Symmetric Functions. Retrieved December 2, 2019 from: https://d31kydh6n6r5j5.cloudfront.net/uploads/sites/66/2019/04/eggecompsdescription.pdf