Power sum symmetric polynomial

In mathematics, specifically in commutative algebra, the power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients can be expressed as a sum and difference of products of power sum symmetric polynomials with rational coefficients. However, not every symmetric polynomial with integral coefficients is generated by integral combinations of products of power-sum polynomials: they are a generating set over the rationals, but not over the integers.

The power sum symmetric polynomial of degree k in ${\displaystyle n}$ variables x₁, ..., x_n, written p_k for k = 0, 1, 2, ..., is the sum of all kth powers of the variables. Formally,

The first few of these polynomials are

Thus, for each nonnegative integer ${\displaystyle k}$, there exists exactly one power sum symmetric polynomial of degree ${\displaystyle k}$ in ${\displaystyle n}$ variables.

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