In mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists...

mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: A monomial, also called...

\{x^{n}\mid n\in \mathbb {N} \}.} This basis is used in Taylor series, amongst others. The monomial basis also forms a basis for the vector space of polynomials...

sequence of monomials is finite. Although Gröbner basis theory does not depend on a particular choice of an admissible monomial ordering, three monomial orderings...

coefficients in F is an F-vector space. One basis for this space is the monomial basis B, consisting of all monomials: B = { 1 , X , X 2 , … } . {\displaystyle...

of a reductive algebraic group by giving an explicit basis of elements called standard monomials. Many of the results have been extended to Kac–Moody...

property of being a Gröbner basis is always relative to a specific monomial order. Besides respecting multiplication, monomial orders are often required...

of the monomials and is commonly called monomial basis. For matrices M m × n {\displaystyle {\mathcal {M}}_{m\times n}} , the standard basis consists...

using the analogy: the functions sin(nx) and cos(nx) are similar to the monomial basis for polynomials. In the complex case the trigonometric polynomials are...

linear algebra amounting to inversion of a matrix. Using a standard monomial basis for our interpolation polynomial L ( x ) = ∑ j = 0 k x j m j {\textstyle...

sum of Legendre polynomials (an orthonormal basis), but not necessarily as an infinite sum of the monomials x n . {\displaystyle x^{n}.} A different generalisation...

In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field. A toric ideal is an ideal generated...

refers to the standard basis defined by the Kronecker delta. In a polynomial ring, it refers to its standard basis given by the monomials, ( X i ) i {\displaystyle...

irreducible representations of the symmetric group on k letters. The standard monomial basis in a finite-dimensional irreducible representation of the general linear...

standard monomial basis for our interpolation polynomial we get the very complicated Vandermonde matrix. By choosing another basis, the Newton basis, we get...

representation in terms of polynomial coefficients is called a monomial basis (a.k.a. 'polynomial basis'). There are other representations of the elements of GF(pn);...

one of its terms). The identity element of this group is the standard monomial basis e n ( x ) = x n = ∑ k = 0 n δ n , k x k . {\displaystyle e_{n}(x)=x^{n}=\sum...

their homogeneous resultant is the determinant of the matrix over the monomial basis of the linear map ( A , B ) ↦ A P + B Q , {\displaystyle (A,B)\mapsto...

These monomial symmetric polynomials form a vector space basis: every symmetric polynomial P can be written as a linear combination of the monomial symmetric...

{\displaystyle (k\geq 0)} . The monomial basis for A ∗ {\displaystyle A_{*}} then gives another choice of basis for A, called the Milnor basis. The dual to the Steenrod...

dimension n + 1. One possible basis for F[x] is a monomial basis: the coordinates of a polynomial with respect to this basis are its coefficients, and the...

respective degrees 0, 1, …, n. Their matrix on the monomial basis is an upper-triangular matrix U (if the monomials are ordered in increasing degrees), with all...

a monomial form. To find the interpolation polynomial p(x) in the vector space P(n) of polynomials of degree n, we may use the usual monomial basis for...

\operatorname {QSym} _{n}} are the monomial basis { M α } {\displaystyle \{M_{\alpha }\}} and the fundamental basis { F α } {\displaystyle \{F_{\alpha...

as well as satisfying a triangularity property when expanded in the monomial basis. In 2007, Haglund, Haiman and Loehr gave a combinatorial formula for...

if m is odd and Z · m+1 if m is even, where Z is a polynomial with monomial basis in m. Therefore r0=1 if m is odd and r0=–1 if m is even is a solution...

polynomial in which no variable occurs to a power of 2 or higher; that is, each monomial is a constant times a product of distinct variables. For example f(x,y...

coefficients on the monomial basis of the polynomials of the form m φ ( f i ) , {\displaystyle m\varphi (f_{i}),} where m is a monomial of degree d − deg ⁡ (...

monomials. Factor: An expression being multiplied. Linear factor: A factor of degree one. Coefficient: An expression multiplying one of the monomials...

Theorem. Let L be a Lie algebra over K and X a totally ordered basis of L. A canonical monomial over X is a finite sequence (x1, x2 ..., xn) of elements of...