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

$P(X_1, X_2) = X_1^3+ X_2^3-7$
$P (X 1, X 2) = 4 X 1 X 2$
$P (X 1, X 2, X 3) = X 1 X 2 X 3 - 2 X 1 X 2 - 2 X 1 X 3 - 2 X 2 X 3$

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

$P(X_1, X_2) = X_1^3+ X_2^3-7=(X_1+X_2)^3-3X_1X_2(X_1+X_2)-7.$

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Symmetric polynomial

Examples

The building blocks for symmetric polynomials

See also: Symmetric polynomial, Elementary symmetric polynomial, Mathematics, Polynomial ring