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 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.

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Complete homogeneous symmetric polynomial

In mathematics, specifically in algebraic combinatorics and commutative algebra, the complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial can be expressed as a polynomial expression in complete homogeneous symmetric polynomials. The complete homogeneous symmetric polynomial of degree k in n variables X1, ..., Xn, written hk for k = 0, 1, 2, ..., is the sum of all monomials of total degree k in the variables.

Newton's identities

In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one variable, they allow expressing the sums of the k-th powers of all roots of P (counted with their multiplicity) in terms of the coefficients of P, without actually finding those roots. These identities were found by Isaac Newton around 1666, apparently in ignorance of earlier work (1629) by Albert Girard.

Ring of symmetric functions

In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number n of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the representation theory of the symmetric group.

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