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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
Order for the terms of a polynomial
mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all (monic) monomials in a given polynomial
Monomial_order
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
Basis of polynomials consisting of monomials
consists of all monomials. The monomials form a basis because every polynomial may be uniquely written as a finite linear combination of monomials (this is an
Monomial_basis
Element of a basis for a function space
depending on the evaluation of the basis functions at the data points). The monomial basis for the vector space of analytic functions is given by { x n ∣ n
Basis_function
In mathematics, a polynomial with two terms
sum of two terms, each of which is a monomial. It is the simplest kind of a sparse polynomial after the monomials. A toric ideal is an ideal that is generated
Binomial_(polynomial)
Generalised alphabetical order
requires the choice of a monomial order, that is a total order, which is compatible with the monoid structure of the monomials. Here "compatible" means
Lexicographic_order
Type of linear representation of a group
{\displaystyle H} . To define the monomial representation, we first need to introduce the notion of monomial space. A monomial space is a triple ( V , X , (
Monomial_representation
monomial group is solvable. Every supersolvable group and every solvable A-group is a monomial group. Factor groups of monomial groups are monomial,
Monomial_group
Polynomial invariant under variable permutations
polynomials that contain only one type of monomial, with only those copies required to obtain symmetry. Any monomial in X1, ..., Xn can be written as X1α1
Symmetric_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
In commutative algebra, a field of mathematics, the monomial conjecture of Melvin Hochster says the following: Let A be a Noetherian local ring of Krull
Monomial_conjecture
Polynomial with negative exponents
In mathematics, a Laurent polynomial (named after Pierre Alphonse Laurent) in one variable over a field F {\displaystyle \mathbb {F} } is a linear combination
Laurent_polynomial
In algebraic geometry, standard monomial theory describes the sections of a line bundle over a generalized flag variety or Schubert variety of a reductive
Standard_monomial_theory
Matrix with one nonzero entry in each row and column
In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is
Generalized permutation matrix
Generalized_permutation_matrix
Ideal generated by differences of monomials
In algebra, a toric ideal is an ideal generated by differences of two monomials. An affine or projective algebraic variety defined by a toric prime ideal
Toric_ideal
Removal of square roots from denominators
denominator of an algebraic fraction are eliminated. If the denominator is a monomial in some radical, say a x n k , {\displaystyle a{\sqrt[{n}]{x}}^{k},} with
Rationalisation_(mathematics)
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
Four finite groups derived from the Leech lattice
suspected that Co0 was transitive on Λ2, and indeed he found a new matrix, not monomial and not an integer matrix. Let η be the 4-by-4 matrix 1 2 ( 1 − 1 − 1 −
Conway_group
Mathematical function
X_{n}).} Here the "lacunary part" Placunary is defined as the sum of all monomials in P which contain only a proper subset of the n variables X1, ..., Xn
Elementary symmetric polynomial
Elementary_symmetric_polynomial
Attribute of a mathematical function
point corresponding to x {\displaystyle x} . Computing the residue of a monomial ∮ C z k d z {\displaystyle \oint _{C}z^{k}\,dz} makes most residue computations
Residue_(complex_analysis)
rows and columns are indexed by monomials. The entries of the matrix depend on the product of the indexing monomials only (cf. Hankel matrices.) Moment
Moment_matrix
Indian mathematician (1932–2020)
Riemann surface.He also introduced and named the concept called Standard monomial theory. He was a recipient of the Padma Bhushan in 2009, the third highest
C._S._Seshadri
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
Type of mathematical expression
bi- with the Greek poly-. That is, it means a sum of many terms (many monomials). The word polynomial was first used in the 17th century. The x {\displaystyle
Polynomial
Polynomial that has three terms
elementary algebra, a trinomial is a polynomial consisting of three terms or monomials. 3 x + 5 y + 8 z {\displaystyle 3x+5y+8z} with x , y , z {\displaystyle
Trinomial
Gröbner bases for non-commutative algebra
(after George Bergman) is a method for confirming whether a given set of monomials of an algebra forms a k {\displaystyle k} -basis. It is an extension of
Bergman's_diamond_lemma
parameter t and a partition λ. They are Schur functions when t is 0 and monomial symmetric functions when t is 1 and are special cases of Macdonald polynomials
Hall–Littlewood_polynomials
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
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
{\displaystyle \mathbb {N} ^{n}} each encoding the exponents within a monomial, consider the multivariate polynomial f ( x ) = ∑ k c k x a k {\displaystyle
Newton_polytope
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
2D graphic with logarithmic scales on both axes
logarithm, though most commonly base 10 (common logs) are used. Given a monomial equation y = a x k , {\displaystyle y=ax^{k},} taking the logarithm of
Log–log_plot
Performing order of mathematical operations
extends to monomials; thus, sin 3x = sin(3x) and even sin 1/2xy = sin(1/2xy), but sin x + y = sin(x) + y, because x + y is not a monomial. However,
Order_of_operations
Vectors whose components are all 0 except one that is 1
polynomials, the standard basis thus consists of the monomials and is commonly called monomial basis. For matrices Mm×n, the standard basis consists
Standard_basis
Optimization problem
{\displaystyle g_{1},\dots ,g_{p}} are monomials. In the context of geometric programming (unlike standard mathematics), a monomial is a function from R + + n {\displaystyle
Geometric_programming
Field of mathematics using techniques from combinatorics and commutative algebra
Adiprasito. Square-free monomial ideal in a polynomial ring and Stanley–Reisner ring of a simplicial complex. Cohen–Macaulay rings. Monomial ring, closely related
Combinatorial commutative algebra
Combinatorial_commutative_algebra
Matrix with no negative elements
is the non-negative monomial matrices: a non-negative matrix has non-negative inverse if and only if it is a (non-negative) monomial matrix. Note that thus
Nonnegative_matrix
Differential algebra
at least one nonzero monomial that has degree deg ( g ) + deg ( h ) {\displaystyle \deg(g)+\deg(h)} . To find such a monomial, pick the one in g {\displaystyle
Weyl_algebra
Process of extending a representation of a subgroup to the parent group
one dimensional representation is called a monomial representation, because it can be represented as monomial matrices. Some groups have the property that
Induced_representation
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
each of which consists of a coefficient from R multiplied by a monomial, where each monomial is a product of finitely many finite powers of indeterminates
Ring_of_symmetric_functions
Polynomial in combinatorial mathematics
of objects partitions that set into cycles; the cycle index monomial of π is a monomial in variables a1, a2, … that describes the cycle type of this
Cycle_index
Topics referred to by the same term
A power series is an infinite sum of monomials. It may also refer to: IBM ThinkPad Power Series Multiple TV series were titled Power or The Power: Power
Power_series_(disambiguation)
Function returning one of only two values
completeness) The algebraic degree of a function is the order of the highest order monomial in its algebraic normal form Circuit complexity attempts to classify Boolean
Boolean_function
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
Algebraic structure
in J (usual sum of vectors). In particular, the product of two monomials is a monomial whose exponent vector is the sum of the exponent vectors of the
Polynomial_ring
Mathematical concept
degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is
Degree_of_a_polynomial
Audio process
N\omega } . All odd monomial terms x n {\displaystyle x^{n}} generate odd harmonics from n down to the fundamental, and all even monomial terms generate even
Waveshaper
Explicitly describes the universal enveloping algebra of a Lie algebra
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 X which is
Poincaré–Birkhoff–Witt theorem
Poincaré–Birkhoff–Witt_theorem
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
Theorem in algebraic geometry
the monomials of degree d in x 1 , … , x n , {\displaystyle x_{1},\ldots ,x_{n},} and its columns are the vectors of the coefficients on the monomial basis
Main theorem of elimination theory
Main_theorem_of_elimination_theory
Infinite sum of monomials
Infinite sum of monomials
Power_series
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
Topics referred to by the same term
Levenberg–Marquardt algorithm, used to solve non-linear least squares problems Leading monomial Linear Monolithic, a National Semiconductor prefix for integrated circuits;
LM
Measure of a mathematical object studied in the field of algebraic geometry
are replaced with their leading monomials, and if these leading monomials are replaced with their radical (monomials obtained by removing exponents).
Dimension of an algebraic variety
Dimension_of_an_algebraic_variety
Relations between power sums and elementary symmetric functions
if the coefficients of any monomial match. Because no individual monomial involves more than k of the variables, the monomial will survive the substitution
Newton's_identities
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
Number of partitions of an integer
distributive law to the product. This expands the product into a sum of monomials of the form x a 1 x 2 a 2 x 3 a 3 ⋯ {\displaystyle x^{a_{1}}x^{2a_{2}}x^{3a_{3}}\cdots
Partition function (number theory)
Partition_function_(number_theory)
for known or naive algorithms for the two problems, and often they are monomials such as n 2 {\displaystyle n^{2}} . Then A {\displaystyle A} is said to
Fine-grained_reduction
Nonlinear differential operator used to study conformal mappings
also clear from the fact that it is in triangular form for the basis of monomials. A flat pseudogroup Γ is said to be "defined by differential equations"
Schwarzian_derivative
Relation between algebraic varieties and polynomial ideals
any monomial ordering) is 1. The number of the common zeros of the polynomials in a Gröbner basis is strongly related to the number of monomials that
Hilbert's_Nullstellensatz
Function with a multiplicative scaling behaviour
{\displaystyle v_{1}\in V_{1},v_{2}\in V_{2},\ldots ,v_{n}\in V_{n}.} Monomials in n {\displaystyle n} variables define homogeneous functions f : F n
Homogeneous_function
Dimensionality of space at which the character of the phase transition changes
may be written as a sum of terms, each consisting of an integral over a monomial of coordinates x i {\displaystyle x_{i}} and fields ϕ i {\displaystyle
Critical_dimension
Mathematical object
example of a flag complex. 6. Let I {\displaystyle I} be a square-free monomial ideal in a polynomial ring S = K [ x 1 , … , x n ] {\displaystyle S=K[x_{1}
Abstract_simplicial_complex
Mathematical functions which are smooth but not analytic
\psi _{n}(x)=x^{n}\,h(x),\qquad x\in \mathbb {R} ,} which agrees with the monomial xn on [−1,1] and vanishes outside the interval (−2,2). Hence, the k-th
Non-analytic_smooth_function
Special type of Boolean function
of bent functions, such as the homogeneous ones or those arising from a monomial over a finite field, but so far the bent functions have defied all attempts
Bent_function
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
Arithmetic in a field with a finite number of elements
m(x). This representation in terms of polynomial coefficients is called a monomial basis (a.k.a. 'polynomial basis'). There are other representations of the
Finite_field_arithmetic
Theorem in transcendental number theory
product is symmetric, for any τ ∈ S N {\displaystyle \tau \in S_{N}} the monomials x τ ( 1 ) h 1 ⋯ x τ ( N ) h N {\displaystyle x_{\tau (1)}^{h_{1}}\cdots
Lindemann–Weierstrass_theorem
On the number of spanning trees in a graph
After collecting terms and performing all possible cancellations, each monomial in the resulting expression represents a spanning tree consisting of the
Kirchhoff's_theorem
of the Hodge bundle, appearing in its integrand. The other factor is a monomial in the ψ i {\displaystyle \psi _{i}} , the first Chern classes of the n
Lambda_g_conjecture
Course designed to prepare students for calculus
This part of precalculus prepares the student for integration of the monomial x p {\displaystyle x^{p}} in the instance of p = − 1 {\displaystyle p=-1}
Precalculus
English affixes added before a word
microbacillus, microscope mono-, mon- sole, only monogamy, monotone, monosyllabic, monomial, monobrow multi-, mult- many multicultural, multi-storey, multitude neo-
English_prefix
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
Problem of determining whether polynomials are identical
required runtime. A sparse PIT has at most m {\displaystyle m} nonzero monomial terms. A sparse PIT can be deterministically solved in polynomial time
Polynomial_identity_testing
ring R, together with a given basis similar to the basis of standard monomials of the coordinate ring of a Grassmannian. Hodge algebras were introduced
Hodge_algebra
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
Algorithm in computer algebra
in the ring of polynomials over a field with respect to a monomial order and a second monomial order. As its output, it returns a Gröbner basis of the ideal
FGLM_algorithm
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
Approximation of the definite integral of a function
than that of the divisor). Since pn is by assumption orthogonal to all monomials of degree less than n, it must be orthogonal to the quotient q(x). Therefore
Gaussian_quadrature
Sporadic simple group
edge x-y = (1, 5, 122) is type 3; it is fixed by a Co3. This M22 is the monomial, and a maximal, subgroup of a representation of McL. Wilson (2009) (p. 207)
McLaughlin_sporadic_group
Stability criterion in control theory
{\displaystyle K} . The factoring of K {\displaystyle K} and the use of simple monomials means the evaluation of the rational polynomial can be done with vector
Root_locus_analysis
Name list
artist known for his "Sneaky Snitch" series Sir Ector Hector de Maris Monomial characters Hector, minor character in the book series A Series of Unfortunate
Hector_(given_name)
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
Algorithm for polynomial evaluation
{f_{1}(x)}{f_{2}(x)}}=2x^{3}-2x^{2}-x+1-{\frac {4}{2x-1}}.} Evaluation using the monomial form of a degree n {\displaystyle n} polynomial requires at most n {\displaystyle
Horner's_method
Analysis of the dimensions of different physical quantities
well-defined but the right-hand side is not. Similarly, while one can evaluate monomials (xn) of dimensional quantities, one cannot evaluate polynomials of mixed
Dimensional_analysis
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
Mathematical concept in polynomial theory
map between two spaces of the same dimension. Consider the descending monomial bases of these polynomial vector spaces: { ( x e − 1 , 0 ) , ( x e − 2
Resultant
Graph drawing used to study Riemann surfaces
after George Shabat. For example, take p {\displaystyle p} to be the monomial p ( x ) = x d {\displaystyle p(x)=x^{d}} having only one finite critical
Dessin_d'enfant
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
} This formula is a special case of the kth forward difference of the monomial xn evaluated at x = 0: Δ k x n = ∑ j = 0 k ( − 1 ) k − j ( k j ) ( x +
Table_of_Newtonian_series
Persian mathematician and engineer (c. 953 – c. 1029)
studied the algebra of exponents, and was the first to define the rules for monomials like x, x2, x3 and their reciprocals in the cases of multiplication and
Al-Karaji
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
Tool used in probabilistic polynomial identity testing
can multiply all the terms and check whether the coefficient of every monomial is nonzero. However, this can take exponential time in the number of variables
Schwartz–Zippel_lemma
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
Result on gamma function
suppose, without loss of generality, that P {\displaystyle P} contains a monomial term having a non-zero power of one of the indeterminates Y 0 , Y 1 , …
Hölder's_theorem
Mathematical inequality
are nonnegative integers, the a-mean can be equivalently defined via the monomial symmetric polynomial m a ( x 1 , … , x n ) {\displaystyle m_{a}(x_{1},\dots
Muirhead's_inequality
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
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
MONOMIAL
MONOMIAL
MONOMIAL
MONOMIAL
Boy/Male
Arabic, Muslim
Firm; Fixed; The Greatness
Girl/Female
Indian
Peace of the World
Girl/Female
Hindu, Indian, Marathi
Pearl; The Moon
Male
English
Cornish and English form of French Degaré, probably DIGORY means "strayed, lost."Â
Girl/Female
Assamese, Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Sanskrit, Tamil, Telugu
Collection of Lamps; Light; Intelligent; Delicate
Boy/Male
Tamil
Winner
Girl/Female
Danish, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Punjabi, Sanskrit, Sikh, Telugu
Begining; Time
Surname or Lastname
English (Somerset) and German (also Hücker)
English (Somerset) and German (also Hücker) : occupational name for a peddler or other tradesman, Middle English hucker, hukker (an agent derivative of hukken ‘to hawk or trade’), Middle High German hucker.
Boy/Male
Muslim
Beautiful
Boy/Male
French American English Greek Irish
Curly haired.
MONOMIAL
MONOMIAL
MONOMIAL
MONOMIAL
MONOMIAL
n.
A single algebraic expression; that is, an expression unconnected with any other by the sign of addition, substraction, equality, or inequality.
a.
Consisting of but a single term or expression.
n.
A monomial.