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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
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
Element of a basis for a function space
\{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
Basis_function
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 , X 2 , … } . {\displaystyle
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
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
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
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
Polynomials used for interpolation
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
Lagrange_polynomial
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
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
Mathematical concept in polynomial theory
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
Resultant
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
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
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
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
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
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 {\displaystyle \mathbb
Monomial_ideal
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
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)
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
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
Type of polynomial sequence
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
Sheffer_sequence
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
Sequence
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
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
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
Theorem in algebraic geometry
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 (
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
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
Multi particle state space
{\displaystyle B_{\infty }} is isomorphic to a bosonic Fock space. The monomial x 1 n 1 . . . x k n k {\displaystyle x_{1}^{n_{1}}...x_{k}^{n_{k}}} corresponds
Fock_space
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
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
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
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, h k ( X 1 , X 2 , … , X
Complete homogeneous symmetric polynomial
Complete_homogeneous_symmetric_polynomial
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
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
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
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
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
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
Algebraic structure
of a vector space or free module equipped by a specific basis (here the basis of the monomials). Explicitly, let p = ∑ α ∈ I p α X α , q = ∑ β ∈ J q β
Polynomial_ring
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
Type of polynomial used in Numerical Analysis
coefficients or Bézier coefficients. The first few Bernstein basis polynomials from above in monomial form are: b 0 , 0 ( x ) = 1 , b 0 , 1 ( x ) = 1 − 1 x
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
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
Differential algebra
A_{n}} has a basis { q m p n : m , n ≥ 0 } {\displaystyle \{q^{m}p^{n}:m,n\geq 0\}} . Proof By repeating the commutator relations, any monomial can be equated
Weyl_algebra
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
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
Matrix of geometric progressions
n n {\displaystyle x_{1}x_{2}^{2}\cdots x_{n}^{n}} , which is also the monomial that is obtained by taking the first term of all factors in ∏ 0 ≤ i < j
Vandermonde_matrix
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
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
Function of the coefficients of a polynomial that gives information on its roots
+a_{0}.} It follows from what precedes that the exponents in every monomial a 0 i 0 , … , a n i n {\displaystyle a_{0}^{i_{0}},\dots ,a_{n}^{i_{n}}}
Discriminant
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
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
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
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
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
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
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
Polynomial whose nonzero terms all have the same degree
(or free module) R d {\displaystyle R_{d}} is the number of different monomials of degree d in n variables (that is the maximal number of nonzero terms
Homogeneous_polynomial
Algebraic study of differential equations
_{\mu }p\geq \theta _{\mu }q.} Each derivative has an integer tuple, and a monomial order ranks the derivative by ranking the derivative's integer tuple. The
Differential_algebra
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
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
Relations between power sums and elementary symmetric functions
distinct monomials of degree k obtained by multiplying one variable raised to the power i with k − i distinct other variables (this is the monomial symmetric
Newton's_identities
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
Pair of polynomial sequences
)}^{\mp 1}.} An explicit form of the Chebyshev polynomial in terms of monomials x k {\displaystyle \textstyle x^{k}} can be obtained as follows. Letting
Chebyshev_polynomials
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
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
Italian mathematician (born 1952)
Gröbner bases including her discovery of the FGLM algorithm for changing monomial orderings in Gröbner bases, and for her development of the components of
Patrizia_Gianni
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
from the work of Kashiwara and Lusztig on quantum groups and the standard monomial theory of C. S. Seshadri and Lakshmibai, Littelmann's path model associates
Littelmann_path_model
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
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
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
Algorithms for matrix decomposition
time algorithm for solving nonnegative rank factorization if V contains a monomial sub matrix of rank equal to its rank was given by Campbell and Poole in
Non-negative matrix factorization
Non-negative_matrix_factorization
check this for monomials in the ei's. Now, a monomial of even degree commutes with every element. Therefore if either x or y is a monomial of even degree
Polynomial_identity_ring
it has no nil one-sided ideal other than { 0 } {\displaystyle \{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
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
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
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
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
Type of Dirichlet series associated to number field extensions
using possitive integer coefficients are called monomial groups, and Taketa (1930) proved that all monomial group are solvable groups. Moreover, this is
Artin_L-function
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
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
Error-correcting codes used in wireless communication
it's 1, update the code to remove the monomial μ {\textstyle \mu } from the input code and continue to next monomial, in reverse order of their degree. Let's
Reed–Muller_code
Set of polynomials where any two are orthogonal to each other
m\neq n~.} In other words, the sequence is obtained from the sequence of monomials 1, x, x2, … by the Gram–Schmidt process with respect to this inner product
Orthogonal_polynomials
Number of subsets of a given size
{\displaystyle {\tbinom {n}{k}}} 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
Algebraic expansion of powers of a binomial
_{k'=0}^{n}{\binom {n}{k'}}x^{k'}y^{n-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 concept
weak compositions of d into n parts. In fact, a basis for the space is given by the set of monomials x 1 d 1 ⋯ x n d n {\displaystyle x_{1}^{d_{1}}\cdots
Composition_(combinatorics)
Branch of mathematics
the monomial, demonstrating its use in calculus. We could also interpret this equation as the first two terms of the Taylor expansion of the monomial. Infinitesimals
Deformation_(mathematics)
Symmetric function invariant of graphs
{\displaystyle \lambda } a partition, let m λ {\displaystyle m_{\lambda }} be the monomial symmetric polynomial associated to λ {\displaystyle \lambda } . Consider
Chromatic_symmetric_function
Algebra used in 2D conformal field theories and string theory
{\displaystyle n} is negative), then we may write the operator product of such a monomial as a normally ordered product of divided power derivatives of fields (here
Vertex_operator_algebra
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
Branch of mathematics
variables. Each variable can be raised to a positive integer power. A monomial is a polynomial with one term while two- and three-term polynomials are
Algebra
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
Mathematical object
{\displaystyle x_{1},x_{2},\ldots ,x_{n},} , that is sums of divided power monomials of the form c x 1 [ i 1 ] x 2 [ i 2 ] ⋯ x n [ i n ] {\displaystyle
Divided_power_structure
Roots of multiple multivariate polynomials
there is a leading monomial of some element of the Gröbner basis which is a pure power of this variable. For this test, the best monomial order (that is the
System of polynomial equations
System_of_polynomial_equations
Mapping from p forms to p-1 forms
and Lie derivative, it suffices to prove the Cartan's magic formula for monomial k {\displaystyle k} -forms. There are only two cases: Case 1: α = a d ξ
Interior_product
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