as a polynomial expression in complete homogeneous symmetric polynomials. The complete homogeneous symmetric polynomial of degree k in n variables X1...

a polynomial. In this context other collections of specific symmetric polynomials, such as complete homogeneous, power sum, and Schur polynomials play...

polynomial can be expressed as a polynomial in elementary symmetric polynomials. That is, any symmetric polynomial P is given by an expression involving...

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...

elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of polynomial irreducible...

types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one...

power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational...

Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric k {\displaystyle...

K[X0, X1, X2, ..., XN] is the polynomial ring in N + 1 variables Xi. The polynomial ring is therefore the homogeneous coordinate ring of the projective...

every polynomial which is homogeneous and symmetric in the roots may be expressed as a quasi-homogeneous polynomial in the elementary symmetric functions...

s_{\mu }h_{r}=\sum _{\lambda }s_{\lambda }} where hr is a complete homogeneous symmetric polynomial and the sum is over all partitions λ obtained from μ by...

Greatest common divisior of two polynomials Symmetric function Homogeneous polynomial Polynomial SOS (sum of squares) Polynomial family Quadratic function Cubic...

proof that the spaces Hℓ are pairwise orthogonal and complete in L2(Sn−1). Every homogeneous polynomial p ∈ Pℓ can be uniquely written in the form p ( x )...

of symmetric functions, as a concise relation between the generating series for elementary, complete and power sums homogeneous symmetric polynomials in...

mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, 4 x 2 + 2 x y...

Flag manifolds can be symmetric spaces. Over the complex numbers, the corresponding flag manifolds are the Hermitian symmetric spaces. Over the real numbers...

For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of...

set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables...

this polynomial may be expanded in a polynomial in s whose coefficients are symmetric polynomials in the xi. By the fundamental theorem of symmetric polynomials...

have Casimir invariants of higher order, which correspond to homogeneous symmetric polynomials of higher order. Suppose that g {\displaystyle {\mathfrak...

_{l=1}^{n}lk_{l}=n.} The formula can be expressed in terms of the complete exponential Bell polynomial of n arguments sl = −(l – 1)! tr(Al) as det ( A ) = ( − 1...

P n {\displaystyle \mathbb {P} ^{n}} of some finite family of homogeneous polynomials that generate a prime ideal, the defining ideal of the variety...

is a polynomial. The number P ( 1 ) {\displaystyle P(1)} is the degree of the algebraic set defined by the ideal, in the case of a homogeneous ideal...

_{i}+j-i})_{i,j}^{r\times r}\right),} where hi are the complete homogeneous symmetric polynomials (with hi understood to be 0 if i is negative). For instance...

in k[x0, ..., xn] be a homogeneous polynomial of degree d. It is not well-defined to evaluate  f  on points in Pn in homogeneous coordinates. However,...

of a preorder is the divides relation "x divides y" between integers, polynomials, or elements of a commutative ring. For example, the divides relation...

S^{-1}:=\left\{s^{-1}:s\in S\right\}.} The closure of every symmetric set in a commutative topological group is symmetric. If S is any subset of a commutative topological...

order (i.e., the number of elements) of the symmetric group Sn is n!. A symmetric polynomial is a polynomial P(X1, X2, ..., Xn) in n variables, such that...

symmetric polynomials. In other words, thinking of ai as formal variables, ck "are" σk. A basic fact on symmetric polynomials is that any symmetric polynomial...

can be expressed (almost everywhere) as an infinite sum of Legendre polynomials (an orthonormal basis), but not necessarily as an infinite sum of the...