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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
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
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
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)
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
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 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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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)
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
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
{\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
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
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
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
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
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
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
German mathematician
Littelmann path model and used it to solve several conjectures in standard monomial theory and other areas. Littelmann was an invited speaker at the International
Peter_Littelmann
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
Iranian mathematician (born 1942)
fractions. This topic later found applications in local cohomology, in the monomial conjecture, and other branches of commutative algebra. Zakeri was born
Hossein_Zakeri
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
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
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
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
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
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
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)
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
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
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
Indian mathematician
Indian mathematician at the University of Hyderabad who developed standard monomial theory in collaboration with his PhD supervisor C. S. Seshadri. Musili
Chitikila_Musili
Topics referred to by the same term
largest sum of exponents (for a multivariate polynomial) in any of its monomials; the multiplicative order, that is, the number of times the polynomial
Order_of_a_polynomial
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
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
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
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
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
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 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
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
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
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
Infinite sum of monomials
Infinite sum of monomials
Power_series
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
Correlation inequality in statistical mechanics
flipping, the correlation of any monomial of the spins is non-negative; and the two point correlation of two monomial of the spins is non-negative. The
Griffiths_inequality
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
French mathematician and quantitative psychology researcher
AF99. Aczél, János; Jean-Claude Falmagne (June 1999). "Consistency of Monomial and Difference Representations of Functions Arising from Empirical Phenomena"
Jean-Claude_Falmagne
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
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)
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
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
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
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
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
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
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
Branch of mathematics
extension of the basis field) if and only if the Gröbner basis for any monomial ordering is reduced to {1}. By means of the Hilbert series, one may compute
Algebraic_geometry
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
Concept in mathematics
cyclic group of order m. As a matrix group, its elements may be realized as monomial matrices whose nonzero elements are mth roots of unity. The group G(m,
Complex_reflection_group
Sporadic simple group
of η is odd. Another representation fixes the vector v = (4,-4,022). A monomial and maximal subgroup includes a representation of M22:2, where any α interchanging
Conway_group_Co2
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
the complete homogeneous symmetric polynomial, that is the sum of all monomials of degree k in the variables x i {\displaystyle x_{i}} , and by e k {\displaystyle
Plethystic_exponential
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
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
Trigonometric_polynomial
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
Mathematical operation on random variables
]}=0.\,} Equivalently, the Wick product can be defined by writing the monomial X1, ..., Xk as a "Wick polynomial": X 1 … X k = ∑ S ⊆ { 1 , … , k } E
Wick_product
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
Result of commutative algebra
_{1}}\prod _{2}^{m}(z_{i}+{\tilde {y}}^{r^{i-1}})^{\alpha _{i}}} is a monomial appearing in the left-hand side of the above equation, with coefficient
Noether_normalization_lemma
Infinite sum approximating a probability distribution in terms of its cumulants
n} . The coefficients of n−m/2 term can be obtained by collecting the monomials of the Bell polynomials corresponding to the integer partitions of m.
Edgeworth_series
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
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
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 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
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
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
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
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
Topics referred to by the same term
vision modeling. See CIE 1931 color space#Color matching functions Common Monomial Factor, the factored form of a polynomial, also known as the greatest common
CMF
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
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
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
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
Specific linear basis (mathematics)
(an orthonormal basis), but not necessarily as an infinite sum of the monomials x n . {\displaystyle x^{n}.} A different generalisation is to pseudo-inner
Orthonormal_basis
MONOMIAL
MONOMIAL
MONOMIAL
MONOMIAL
Biblical
winter; reproach
Girl/Female
Arabic, Muslim
Kind; Smooth; Easy
Boy/Male
Hindu, Indian, Punjabi, Sikh
One who Attains True Peace
Girl/Female
Muslim/Islamic
Purity
Girl/Female
Tamil
Supreeta | ஸà¯à®ªà¯à®°à®¿à®¤à®¾, ஸà¯à®ªà¯à®°à®¿à®¤à®¾Â
Adored one, Beloved, Endearing to all, Well pleased
Girl/Female
Arabic, Muslim
Pleiades
Boy/Male
Gaelic
Of the strange Gauls.
Girl/Female
Arabic, Muslim, Persian
Happy; Pleased
Girl/Female
French
An Old FrenchLatin 'aestimatus' meaning esteemed, or 'amatus' meaning loved.
Girl/Female
Tamil
The princess
MONOMIAL
MONOMIAL
MONOMIAL
MONOMIAL
MONOMIAL
a.
Consisting of but a single term or expression.
n.
A monomial.
n.
A single algebraic expression; that is, an expression unconnected with any other by the sign of addition, substraction, equality, or inequality.