Symmetric Polynomial - Special Kinds of Symmetric Polynomials - Power-sum Symmetric Polynomials

Power-sum Symmetric Polynomials

For each integer k ≥ 1, the monomial symmetric polynomial m(k,0,…,0)(X1, …, Xn) is of special interest, and called the power sum symmetric polynomial pk(X1, …, Xn), so

All symmetric polynomials can be obtained from the first n power sum symmetric polynomials by additions and multiplications, possibly involving rational coefficients. More precisely,

Any symmetric polynomial in X1, …, Xn can be expressed as a polynomial expression with rational coefficients in the power sum symmetric polynomials p1(X1, …, Xn), …, pn(X1, …, Xn).

In particular, the remaining power sum polynomials pk(X1, …, Xn) for k > n can be so expressed in the first n power sum polynomials; for example


In contrast to the situation for the elementary and complete homogeneous polynomials, a symmetric polynomial in n variables with integral coefficients need not be a polynomial function with integral coefficients of the power sum symmetric polynomials. For an example, for n = 2, the symmetric polynomial

has the expression

Using three variables one gets a different expression

\begin{align}m_{(2,1)}(X_1,X_2,X_3) &= X_1^2 X_2 + X_1 X_2^2 + X_1^2 X_3 + X_1 X_3^2 + X_2^2 X_3 + X_2 X_3^2\\ &= p_1(X_1,X_2,X_3)p_2(X_1,X_2,X_3)-p_3(X_1,X_2,X_3). \end{align}

The corresponding expression was valid for two variables as well (it suffices to set X3 to zero), but since it involves p3, it could not be used to illustrate the statement for n = 2. The example shows that whether or not the expression for a given monomial symmetric polynomial in terms of the first n power sum polynomials involves rational coefficients may depend on n. But rational coefficients are always needed to express elementary symmetric polynomials (except the constant ones, and e1 which coincides with the first power sum) in terms of power sum polynomials. The Newton identities provide an explicit method to do this; it involves division by integers up to n, which explains the rational coefficients. Because of these divisions, the mentioned statement fails in general when coefficients are taken in a field of finite characteristic; however it is valid with coefficients in any ring containing the rational numbers.

Read more about this topic:  Symmetric Polynomial, Special Kinds of Symmetric Polynomials