Symmetric polynomial

In mathematics, a symmetric polynomial is a polynomial in n variables $P(X_{1},X_{2},...,X_{n})$, such that if some of the variables get interchanged, the polynomial stays the same.

Examples

• $P(X_{1},X_{2})=X_{1}^{3}+X_{2}^{3}-7$
• $P(X_{1},X_{2})=4X_{1}X_{2}$
• $P(X_{1},X_{2},X_{3})=X_{1}X_{2}X_{3}-2X_{1}X_{2}-2X_{1}X_{3}-2X_{2}X_{3}$

are all symmetric. The polynomial $P(X_{1},X_{2})=X_{1}-X_{2}$ is not symmetric, since if we exchange $X_{1}$ and $X_{2}$ we get the polynomial $X_{2}-X_{1}$ which is not the same thing.

The building blocks for symmetric polynomials

For each n, there exist n so-called elementary symmetric polynomials in $X_{1},X_{2},...,X_{n}$. They are the building blocks for all symmetric polynomials in these variables, meaning that any symmetric polynomial in n variables can be obtained from the elementary symmetric polynomials via several multiplications and additions. More precisely: any symmetric polynomial in n variables is a polynomial of the n elementary symmetric polynomials in these variables. For example, for n=2, there are only two elementary symmetric polynomials, $X_{1}+X_{2}$ and $X_{1}X_{2}$. The first polynomial in the list of examples above can then be written as

$P(X_{1},X_{2})=X_{1}^{3}+X_{2}^{3}-7=(X_{1}+X_{2})^{3}-3X_{1}X_{2}(X_{1}+X_{2})-7.$