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. Formally,
The formula can also be written as:
Indeed, lp is just the multiplicity of p in the sequence ik.
The first few of these polynomials are
Thus, for each nonnegative integer k, there exists exactly one complete homogeneous symmetric polynomial of degree k in n variables.
Another way of rewriting the definition is to take summation over all sequences ik, without condition of ordering ip ≤ ip + 1:
here mp is the multiplicity of number p in the sequence ik.
For example
The polynomial ring formed by taking all integral linear combinations of products of the complete homogeneous symmetric polynomials is a commutative ring.
The following lists the n basic (as explained below) complete homogeneous symmetric polynomials for the first three positive values of n.
For n = 1:
For n = 2:
For n = 3:
The complete homogeneous symmetric polynomials are characterized by the following identity of formal power series in t:
(this is called the generating function, or generating series, for the complete homogeneous symmetric polynomials). Here each fraction in the final expression is the usual way to represent the formal geometric series that is a factor in the middle expression. The identity can be justified by considering how the product of those geometric series is formed: each factor in the product is obtained by multiplying together one term chosen from each geometric series, and every monomial in the variables Xi is obtained for exactly one such choice of terms, and comes multiplied by a power of t equal to the degree of the monomial.
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.
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.
En mathématiques, et plus particulièrement en algèbre, les identités de Newton (connues également sous le nom de formules de Newton-Girard) sont des relations entre deux types de polynômes symétriques, les polynômes symétriques élémentaires, et les sommes de Newton, c'est-à-dire les sommes de puissances des indéterminées. Évaluées aux racines d'un polynôme P à une variable, ces identités permettent d'exprimer les sommes des k-ièmes puissances de toutes les racines de P (comptées avec leur multiplicité) en fonction des coefficients de P, sans qu'il soit nécessaire de déterminer ces racines.
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.
Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics a
The purpose of this teaching lab is to put together all the concepts learned during the course into electrical energy by the implementation of an islanded production unit. The number of places is limi
Continuum conservation laws (e.g. mass, momentum and energy) will be introduced. Mathematical tools, including basic algebra and calculus of vectors and Cartesian tensors will be taught. Stress and de
We prove an identity relating the permanent of a rank 2 matrix and the determinants of its Hadamard powers. When viewed in the right way, the resulting formula looks strikingly similar to an identity of Carlitz and Levine, suggesting the possibility that t ...
A key challenge across many disciplines is to extract meaningful information from data which is often obscured by noise. These datasets are typically represented as large matrices. Given the current trend of ever-increasing data volumes, with datasets grow ...
We consider two problems regarding arithmetic progressions in symmetric sets in the finite field (product space) model. First, we show that a symmetric set S subset of Z(q)(n) containing vertical bar S vertical bar = mu . q(n) elements must contain at leas ...