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Basis of polynomials consisting of monomials
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
Monomial_basis
Element of a basis for a function space
interpolated. Basis functions are also used extensively in finite element methods, splines, wavelets, and other approximation methods. The monomial functions
Basis_function
Polynomial with only one term
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
Monomial
Mathematical construct in computer algebra
sequence of monomials is finite. Although Gröbner basis theory does not depend on a particular choice of an admissible monomial ordering, three monomial orderings
Gröbner_basis
Set of vectors used to define coordinates
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,X2,…}. Any set of polynomials
Basis_(linear_algebra)
Vectors whose components are all 0 except one that is 1
standard basis thus consists of the monomials and is commonly called monomial basis. For matrices Mm×n, the standard basis consists of the m×n–matrices with
Standard_basis
Order for the terms of a polynomial
property of being a Gröbner basis is always relative to a specific monomial order. Besides respecting multiplication, monomial orders are often required
Monomial_order
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
Standard_monomial_theory
Algebra in algebraic topology
{\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
Steenrod_algebra
Polynomial interpolation using derivative values
generalized Lagrange basis. Other notable common bases are the Newton basis of Newton's divided difference method (see here), the monomial basis of the confluent
Hermite_interpolation
Specific linear basis (mathematics)
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
Orthonormal_basis
Concept in mathematics
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
Trigonometric_polynomial
Mathematical concept in polynomial theory
their homogeneous resultant is the determinant of the matrix over the monomial basis of the linear map (A,B)↦AP+BQ, where A runs over the bivariate homogeneous
Resultant
Polynomials used for interpolation
linear algebra amounting to inversion of a matrix. Using a standard monomial basis for our interpolation polynomial L(x)=∑j=0kxjmj, we must invert the
Lagrange_polynomial
Orthogonal symmetric polynomial family
as well as satisfying a triangularity property when expanded in the monomial basis. In 2007, Haglund, Haiman and Loehr gave a combinatorial formula for
Macdonald_polynomials
Form of interpolation
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
Polynomial_interpolation
Type of polynomial sequence
one of its terms). The identity element of this group is the standard monomial basis en(x)=xn=∑k=0nδn,kxk. Two important subgroups are the group of Appell
Sheffer_sequence
In mathematics, a polynomial with two terms
For every admissible monomial ordering, the minimal Gröbner basis of a toric ideal consists only of differences of monomials. (This is an immediate
Binomial_(polynomial)
Mathematical expression
standard monomial basis for our interpolation polynomial we get the very complicated Vandermonde matrix. By choosing another basis, the Newton basis, we get
Newton_polynomial
Finite or infinite ordered list of elements
of a sequence can be functions instead of numbers. For example, the monomial basis for polynomials of a single variable forms the sequence (x↦1,x↦x,x↦x2
Sequence
Group of symmetries of an n-dimensional hypercube
Houyi (2024), "The weak order on the hyperoctahedral group and the monomial basis for the Hopf algebra of signed permutations", Discrete Mathematics,
Hyperoctahedral_group
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
Examples_of_vector_spaces
Ideal generated by one-term polynomials
In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field. Let 𝕂 be a field and R=𝕂[x]
Monomial_ideal
Combinatorial object in representation theory
irreducible representations of the symmetric group on k letters. The standard monomial basis in a finite-dimensional irreducible representation of the general linear
Young_tableau
Polynomial invariant under variable permutations
These monomial symmetric polynomials form a vector space basis: every symmetric polynomial P can be written as a linear combination of the monomial symmetric
Symmetric_polynomial
Irish mathematician (1944–2024)
Funct. Anal. 237 (2006), no. 1, 338–349. Dineen, Seán; Mujica, Jorge "A monomial basis for the holomorphic functions on $c_0$". Proc. Amer. Math. Soc. 141
Seán_Dineen
Multiplicative factor in a mathematical expression
multivariate polynomials with respect to a monomial order, see Gröbner basis § Leading term, coefficient and monomial. In linear algebra, a system of linear
Coefficient
Arithmetic in a field with a finite number of elements
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);
Finite_field_arithmetic
Multi particle state space
and Bargmann showed that B∞ is isomorphic to a bosonic Fock space. The monomial x1n1...xknk corresponds to the Fock state |n0,n1,…,nk⟩ν=|ψ0⟩n0|ψ1⟩n1⋯|ψk⟩nk
Fock_space
Skeletonized version of algebraic geometry
that can be expressed as the tropical sum of a finite number of monomial terms. A monomial term is a tropical product (and/or quotient) of a constant and
Tropical_geometry
Basis of a type of algebraic structure
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
Canonical_basis
Mathematical puzzle
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
The_monkey_and_the_coconuts
x_{i_{k}}^{\alpha _{k}}.\,} (Note that the elements of the monomial basis may not be monomials.) The fundamental basis consists F 0 = 1 {\displaystyle F_{0}=1} and
Quasisymmetric_function
Gröbner bases for non-commutative algebra
method for confirming whether a given set of monomials of an algebra forms a k {\displaystyle k} -basis. It is an extension of Gröbner bases to non-commutative
Bergman's_diamond_lemma
Type of polynomial used in Numerical Analysis
coefficients or Bézier coefficients. The first few Bernstein basis polynomials from above in monomial form are: b0,0(x)=1 ,b0,1(x)=1−1x ,b1,1(x)=0+1xb0,2(x)=1−2x+1x2
Bernstein_polynomial
Explicitly describes the universal enveloping algebra of a Lie algebra
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
Poincaré–Birkhoff–Witt theorem
Poincaré–Birkhoff–Witt_theorem
Root-finding algorithm for polynomials
α ) {\displaystyle (X-\alpha )} is a linear factor of P(X). In the monomial basis the linear map M X {\displaystyle M_{X}} is represented by a companion
Jenkins–Traub_algorithm
Theorem in algebraic geometry
rows are indexed by the monomials of degree d in x1,…,xn, and its columns are the vectors of the coefficients on the monomial basis of the polynomials of
Main theorem of elimination theory
Main_theorem_of_elimination_theory
free module over some ring R, together with a given basis similar to the basis of standard monomials of the coordinate ring of a Grassmannian. Hodge algebras
Hodge_algebra
Set of matrices
coefficients. For instance, Littlewood polynomials have coefficients ±1 in the monomial basis. Researchers such as Kurt Mahler, Andrew Odlyzko, Bjorn Poonen and Peter
Bohemian_matrices
Expression in commutative algebra
variables X1, ..., Xn, written hk for k = 0, 1, 2, ..., is the sum of all monomials of total degree k in the variables. Formally, hk(X1,X2,…,Xn)=∑1≤i1≤i2≤⋯≤ik≤nXi1Xi2⋯Xik
Complete homogeneous symmetric polynomial
Complete_homogeneous_symmetric_polynomial
Measure of a mathematical object studied in the field of algebraic geometry
of the Gröbner basis are replaced with their leading monomials, and if these leading monomials are replaced with their radical (monomials obtained by removing
Dimension of an algebraic variety
Dimension_of_an_algebraic_variety
Type of polynomial
variable occurs to a power of 2 {\displaystyle 2} or higher; that is, each monomial is a constant times a product of distinct variables. For example f ( x
Multilinear_polynomial
Relation between algebraic varieties and polynomial ideals
its reduced Gröbner basis (for any monomial ordering) is 1. The number of the common zeros of the polynomials in a Gröbner basis is strongly related to
Hilbert's_Nullstellensatz
Algorithm for computing Gröbner bases
Output A Gröbner basis G for I G := F For every fi, fj in G, denote by gi the leading term of fi with respect to the given monomial ordering, and by aij
Buchberger's_algorithm
an R-linear combination of monomial symmetric functions, and the distinct monomial symmetric functions therefore form a basis of ΛR as an R-module. The
Ring_of_symmetric_functions
Algebraic structure
of a vector space or free module equipped by a specific basis (here the basis of the monomials). Explicitly, let p=∑α∈IpαXα,q=∑β∈JqβXβ, where I and J are
Polynomial_ring
Boolean polynomials as sums of monomials
Zhegalkin monomials, with the empty set denoted by 0. A given monomial's presence or absence in a polynomial corresponds to that monomial's coefficient
Algebraic_normal_form
Algorithm in computer algebra
a Gröbner basis of a zero-dimensional ideal in the ring of polynomials over a field with respect to a monomial order and a second monomial order. As its
FGLM_algorithm
Algorithms for computing Gröbner bases
(f1) + Gprev then we will construct matrices whose rows are m f1 such that m is a monomial not divisible by the leading term of an element of Gprev. This strategy
Faugère's F4 and F5 algorithms
Faugère's_F4_and_F5_algorithms
On polynomial rings over fields
polynomials. If the degree of these polynomials is bounded, the number of their monomials is also bounded. Expressing that one has a syzygy provides a system of
Hilbert's_syzygy_theorem
Polynomial with 1 as leading coefficient
a monomial order is generally fixed. In this case, a polynomial may be said to be monic if it has 1 as its leading coefficient (in the monomial order)
Monic_polynomial
Algebraic expansion of powers of a binomial
(x+y)n=∑k=0n(nk)xn−kyk=∑k′=0n(nk′)xk′yn−k′, and the coefficient of the same monomial in the left and right-hand side expressions of the 2nd equality must be
Binomial_theorem
Mathematical notation
product Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and contravariance
Multi-index_notation
Error-correcting codes used in wireless communication
contains any number of errors up to 2m−r−1−1 included. If you consider a monomial μ of the highest degree d in px and sum all the evaluation points of the
Reed–Muller_code
Statistics concept
individual coefficients in a polynomial regression fit, since the underlying monomials can be highly correlated. For example, x and x2 have correlation around
Polynomial_regression
Differential algebra
statement, f has no monomial pnqm with n>0 or m>0. Theorem—An has a basis {qmpn:m,n≥0}. Proof By repeating the commutator relations, any monomial can be equated
Weyl_algebra
Function of a polynomial's coefficients that gives information about its roots
P=anxn+an−1xn−1+⋯+a0. It follows from what precedes that the exponents in every monomial a0i0,…,anin appearing in the discriminant satisfy the two equations i0+i1+⋯+in=2n−2
Discriminant
Branch of mathematics
(over an algebraically closed extension of the basis field) if and only if the Gröbner basis for any monomial ordering is reduced to {1}. By means of the
Algebraic_geometry
In linear algebra, generated subspace
0, 0)} is the intersection of all of these vector spaces. The set of monomials xn, where n is a non-negative integer, spans the space of polynomials
Linear_span
Type of symmetric polynomials in mathematics
that page). Schur polynomials can be expressed as linear combinations of monomial symmetric functions mμ with non-negative integer coefficients Kλμ called
Schur_polynomial
Polynomial in numerical analysis
expresses the polynomial in a particular basis, namely that of the monomials. If the polynomial is expressed in another basis, then the problem of finding its
Wilkinson's_polynomial
Method to evaluate polynomials in Bernstein form
Bézier curve De Boor's algorithm Horner scheme to evaluate polynomials in monomial form Clenshaw algorithm to evaluate polynomials in Chebyshev form Delgado
De_Casteljau's_algorithm
Concept in stochastic analysis
and comparing paths. These iterated integrals play a role similar to monomials in a Taylor expansion: they provide a coordinate system that captures
Rough_path
Matrix of geometric progressions
the product of the diagonal entries of V is x1x22⋯xnn, which is also the monomial that is obtained by taking the first term of all factors in ∏0≤i<j≤n(xj−xi)
Vandermonde_matrix
Type of function
function on the interval with its Fourier series. If one begins with the monomial sequence { 1 , x , x 2 , … } {\displaystyle \left\{1,x,x^{2},\dots \right\}}
Orthogonal_functions
Geometric arrangement of a nodal group
solving a linear system, or by enforcing that the stencil is exact for monomials up to the degree of the stencil. For equi-spaced nodes, they may be calculated
Stencil_(numerical_analysis)
Infinite sum that is considered independently from any notion of convergence
defined similarly by replacing the powers of a single indeterminate by monomials in several indeterminates. Formal power series are widely used in combinatorics
Formal_power_series
Algorithm for solving systems of linear equations
polynomial equations. This generalization depends heavily on the notion of a monomial order. The choice of an ordering on the variables is already implicit in
Gaussian_elimination
Topics referred to by the same term
set of generators for a polynomial ideal into a Gröbner basis with respect to some monomial order This disambiguation page lists articles associated
Buchberger
Mathematical ring
ring, is a quotient of a polynomial algebra over a field by a square-free monomial ideal. Such ideals are described more geometrically in terms of finite
Stanley–Reisner_ring
Pair of polynomial sequences
(x+x2−1 )±1=(x−x2−1 )∓1. An explicit form of the Chebyshev polynomial in terms of monomials xk can be obtained as follows. Letting ℜ denote the real part of a complex
Chebyshev_polynomials
Polynomial whose nonzero terms all have the same degree
dimension of the vector space (or free module) Rd is the number of different monomials of degree d in n variables (that is the maximal number of nonzero terms
Homogeneous_polynomial
String that is strictly smaller in lexicographic order than all of its rotations
with the "noncommutative monomials" (i.e., products of the xa) in R; namely, we identify a word (a1,a2,...,an) with the monomial xa1xa2...xan. Thus, the
Lyndon_word
Non-tensorial representation of the spin group
uniquely to an algebra homomorphism Cℓ(V, g) → Mat(2k, ℂ) by sending the monomial eμ1 ⋅⋅⋅ eμk in the Clifford algebra to the product γμ1 ⋅⋅⋅ γμk of matrices
Spinor
monomials. Factor: An expression being multiplied. Linear factor: A factor of degree one. Coefficient: An expression multiplying one of the monomials
List_of_polynomial_topics
Mathematical study of invariants under symmetries
given by the theory of standard monomials. Simple examples of invariant theory come from computing the invariant monomials from a group action. For example
Invariant_theory
that the monomials Sn,Sn−1T,Sn−2T2,⋯,Tn, are just one possible basis for the space of degree n homogeneous polynomials. In fact, any basis will do. This
Rational_normal_curve
Algebraic study of differential equations
∀p,q∈ΘY, ∀θμ∈Θ:p≥q⇒θμp≥θμq. Each derivative has an integer tuple, and a monomial order ranks the derivative by ranking the derivative's integer tuple. The
Differential_algebra
Algebra over a field where binary multiplication is not necessarily associative
with basis consisting of all non-associative monomials, finite formal products of elements of X retaining parentheses. The product of monomials u, v is
Non-associative_algebra
Class of error-correcting code
equivalent. In more generality, if there is an n × n {\displaystyle n\times n} monomial matrix M : F q n → F q n {\displaystyle M\colon \mathbb {F} _{q}^{n}\to
Linear_code
American mathematician (born 1937)
toroidal embedding theory, the geometric approach to varieties defined by monomials. With Dave Bayer he published a paper "What can be computed in algebraic
David_Mumford
English affixes added before a word
recognizable and relating to present-day idioms) English (that is, "native") basis. Conceptualized thus, anglicized neo-classical English words such as deceive
English_prefix
ideal other than {0}, then it has no nil one-sided ideal other than {0}. Monomial conjecture on Noetherian local rings Existence of perfect cuboids and associated
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Characteristic classes of vector bundles
its tangent bundle. If M is also compact and of dimension 2d, then each monomial of total degree 2d in the Chern classes can be paired with the fundamental
Chern_class
Theorem in transcendental number theory
eγ(N)) by assumption. Since the product is symmetric, for any τ∈SN the monomials xτ(1)h1⋯xτ(N)hN and x1h1⋯xNhN have the same coefficient in the expansion
Lindemann–Weierstrass_theorem
Group in group theory and physics
Poincaré–Birkhoff–Witt theorem, it is thus the free vector space generated by the monomials zjp1k1p2k2⋯pnknq1ℓ1q2ℓ2⋯qnℓn , where the exponents are all non-negative
Heisenberg_group
Index of articles associated with the same name
sum converges. In the study of power series, a sum of infinitely many monomials with distinct positive integer exponents, again considered as an abstract
Formal_sum
Mathematical concept
=\delta _{\mu \lambda }} . Here, m λ {\displaystyle m_{\lambda }} is a monomial symmetric function and h μ {\displaystyle h_{\mu }} is a product of completely
Frobenius_characteristic_map
Number of subsets of a given size
the binomial coefficient (nk) can be defined as the coefficient of the monomial Xk in the expansion of (1 + X)n. The same coefficient also occurs (if k
Binomial_coefficient
Type of Dirichlet series associated to number field extensions
using positive integer coefficients are called monomial groups, and Taketa (1930) proved that all monomial group are solvable groups. Moreover, this is
Artin_L-function
Mathematical object in abstract algebra
is easily described as k[x,x−1]/xk[x]. This module has a basis consisting of "inverse monomials", that is x−n for n = 0, 1, 2, …. Multiplication by scalars
Injective_module
Association of cohomology classes to principal bundles
class. The number of distinct characteristic numbers is the number of monomials of degree n in the characteristic classes, or equivalently the partitions
Characteristic_class
Tool in mathematical dimension theory
Gröbner basis of I for a monomial ordering refining the total degree partial ordering and G the (homogeneous) ideal generated by the leading monomials of the
Hilbert series and Hilbert polynomial
Hilbert_series_and_Hilbert_polynomial
System of complete and orthogonal polynomials
from the recursion formula, expresses the Legendre polynomials by simple monomials and involves the generalized form of the binomial coefficient. The reversal
Legendre_polynomials
the paths form a basis for the root operators like that of a monomial representation: when a root operator is applied to the basis element for a path
Littelmann_path_model
is odd or rankα≤1. The rank of a k-form α means the minimum number of monomial terms (exterior products of one-forms) that must be summed to produce α
Exterior_calculus_identities
where x{m} is a vector containing a basis for the space of forms of degree m in x (such as all monomials of degree m), H is any symmetric matrix satisfying
Polynomial_SOS
Mathematical object studied in the field of algebraic geometry
variables (not always needed); then a Gröbner basis computation for another monomial ordering to compute the projection and to prove that it is generically
Algebraic_variety
Mathematical expression
area is to quantify the mathematical coincidence idea; for example, for monomials in several real numbers, take the logarithmic form and consider how small
Continued_fraction
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MONOMIAL BASIS
MONOMIAL BASIS
MONOMIAL BASIS
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