Symmetric Polynomials
A symmetric polynomial \( P(x, y, z, \ldots) \) is a simplified polynomial - meaning all like terms have been combined - that remains unchanged when any pair of variables, such as \( x \) and \( y \), are interchanged.
Put simply, a symmetric polynomial is an algebraic expression in two or more variables that remains identical under the exchange of any two of those variables.
For instance, take the polynomial $ 2a + 2b + c $. Swapping $ a $ and $ b $ yields $ 2b + 2a + c $, which is algebraically equivalent to the original expression. This is a straightforward example of a symmetric polynomial.
In symmetric polynomials, the order of the variables is irrelevant.
How can you determine whether a polynomial is symmetric? To verify whether a polynomial is symmetric, select any two variables and interchange them. Rewrite the resulting polynomial and compare it to the original expression. If they are identical, the polynomial is symmetric; otherwise, it is not. Symmetric polynomials are foundational in the study of symmetric functions.
Worked Example
Consider the polynomial:
\[ 2x^2 - 3xy + 2y^2 \]
By definition, a polynomial is symmetric if it remains invariant under the exchange of any two variables, in this case \( x \) and \( y \).
Interchanging \( x \) and \( y \), we obtain:
\[ 2y^2 - 3yx + 2x^2 \]
Now rewrite each term with variables in standard alphabetical order:
\[ 2x^2 - 3xy + 2y^2 \]
This matches the original polynomial exactly, confirming that it is symmetric.
Example 2
Now consider the following polynomial:
\[ 3xy - 2xz + 3yz \]
Swap \( x \) and \( y \):
\[ 3yx - 2yz + 3xz \]
Rewriting with standard variable ordering within each term gives:
\[ 3xy - 2yz + 3xz \]
This is clearly different from the original polynomial, which included the term \(-2xz\), now replaced by \(-2yz\).
Therefore, this polynomial is not symmetric.
Remarks
Here are some additional observations regarding symmetric polynomials:
- A polynomial composed entirely of pure powers is symmetric, since each variable appears in the same form: \[ x^2 + y^2 + z^2 \]
- A polynomial that includes all possible cross-product terms is also symmetric: \[ xy + yz + zx \]
And so forth.
