Elementary symmetric polynomial

In mathematics, elementary symmetric polynomials are basic building block for symmetric polynomials.

Consider the variables $A, X 1, X 2, X 3$ . We have that

(A + X 1)(A + X 2)(A + X 3) = A 3 + (X 1 + X 2 + X 3) A 2 + (X 1 X 2 + X 2 X 3 + X 1 X 3) A + X 1 X 2 X 3 .

The coefficients of the powers of $A$

$X_1 + X_2 + X_3, \ X_1 X_2 + X_2 X_3 + X_1 X_3, \ X_1 X_2 X_3$

are the elementary symmetric polynomials in 3 variables. Note that these polynomials are indeed symmetric, as when some variables are interchanged, the polynomials stay the same.

In the same way, one can write

$(A+X_1)(A+X_2)\cdots(A+X_n)= A^n+ (X_1 + X_2 +\cdots+ X_n)A^{n-1}+\cdots+X_1 X_2\cdots X_n$

and the obtained coefficients of the powers of $A$ are the $n$ elementary symmetric polynomials in $n$ variables.

Notice that for each $k$ between 1 and $n$ , there exists exactly one elementary symmetric polynomial of degree $k$ .

The uses of these polynomials are described in the symmetric polynomials article.

Categories: Abstract algebra

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01-04-2007 01:21:04

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