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Concept
Power sum symmetric polynomial
Summary
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 variables x1, ..., xn, written pk 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 , there exists exactly one power sum symmetric polynomial of degree in variables.
The polynomial ring formed by taking all integral linear combinations of products of the power sum symmetric polynomials is a commutative ring.
The following lists the power sum symmetric polynomials of positive degrees up to n for the first three positive values of In every case, is one of the polynomials. The list goes up to degree n because the power sum symmetric polynomials of degrees 1 to n are basic in the sense of the theorem stated below.
For n = 1:
For n = 2:
For n = 3:
The set of power sum symmetric polynomials of degrees 1, 2, ..., n in n variables generates the ring of symmetric polynomials in n variables. More specifically:
Theorem. The ring of symmetric polynomials with rational coefficients equals the rational polynomial ring The same is true if the coefficients are taken in any field of characteristic 0.
However, this is not true if the coefficients must be integers. For example, for n = 2, the symmetric polynomial
has the expression
which involves fractions. According to the theorem this is the only way to represent in terms of p1 and p2.
Official source
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