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

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

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

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...

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

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

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

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

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...

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

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...

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

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...

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...

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

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...

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

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 )...

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

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...

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,...

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...

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...

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

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

stroke Sole sufficient operator Symmetric Boolean function Symmetric difference Zhegalkin polynomial Boolean domain Complete Boolean algebra Interior algebra...

as a change of affine coordinates may map indeterminates on non-homogeneous polynomials. Affine spaces over topological fields, such as the real or the...

_{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...

the degree of polynomials. The projective Nullstellensatz states that, for any homogeneous ideal I that does not contain all polynomials of a certain degree...

called complete. Roughly, a vector space is complete provided that it contains all necessary limits. For example, the vector space of polynomials on the...