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Type of polynomial
In algebra, a multilinear polynomial is a multivariate polynomial that is linear (meaning affine) in each of its variables separately, but not necessarily
Multilinear_polynomial
Polynomial whose nonzero terms all have the same degree
Multi-homogeneous polynomial Quasi-homogeneous polynomial Diagonal form Graded algebra Hilbert series and Hilbert polynomial Multilinear form Multilinear map Polarization
Homogeneous_polynomial
Map from multiple vectors to an underlying field of scalars, linear in each argument
In abstract algebra and multilinear algebra, a multilinear form on a vector space V {\displaystyle V} over a field K {\displaystyle K} is a map f : V k
Multilinear_form
Algebraic structure
mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by k[V]
Ring_of_polynomial_functions
Tool used in probabilistic polynomial identity testing
diagrams). A read-once branching program can be represented by a multilinear polynomial which computes (over any field) on {0,1}-inputs the same Boolean
Schwartz–Zippel_lemma
Study of Boolean functions via discrete Fourier analysis
f\colon \{-1,1\}^{n}\to \mathbb {R} } has a unique expansion as a multilinear polynomial: f ( x ) = ∑ S ⊆ [ n ] f ^ ( S ) χ S ( x ) , χ S ( x ) = ∏ i ∈ S
Analysis_of_Boolean_functions
Function returning one of only two values
a multilinear polynomial in R n {\displaystyle \mathbb {R} ^{n}} , constructed by summing the truth table values multiplied by indicator polynomials: f
Boolean_function
Theorem about complexity measures of Boolean functions
expressed in a unique way as a multilinear polynomial. The degree of f {\displaystyle f} is the degree of this unique polynomial, denoted deg ( f ) {\displaystyle
Sensitivity_theorem
Polynomial whose Laplacian is zero
spherical harmonics Multilinear polynomial Walsh, J. L. (1927). "On the Expansion of Harmonic Functions in Terms of Harmonic Polynomials". Proceedings of
Harmonic_polynomial
is multilinear. If a ring is finitely generated by n elements as a module over its center then it satisfies every alternating multilinear polynomial of
Polynomial_identity_ring
Interpolation on functions of more than one variable
n-linear interpolation (see bi- and trilinear interpolation and multilinear polynomial) n-cubic interpolation (see bi- and tricubic interpolation) Kriging
Multivariate_interpolation
American mathematician
S2CID 17827200. Guth, Larry (2010), "The endpoint case of the Bennett–Carbery–Tao multilinear Kakeya conjecture", Acta Mathematica, 205 (2): 263–286, arXiv:0811.2251
Larry_Guth
Method of interpolating functions on a 2D grid
way is to write the solution to the interpolation problem as a multilinear polynomial f ( x , y ) ≈ a 00 + a 10 x + a 01 y + a 11 x y , {\displaystyle
Bilinear_interpolation
Australian and American mathematician (born 1975)
[T04b] With Camil Muscalu and Christoph Thiele, Tao considered certain multilinear singular integral operators with the multiplier allowed to degenerate
Terence_Tao
Geometric transformation that preserves lines but not angles nor the origin
affine transformations Bent function Flat (geometry) Homography Multilinear polynomial Berger 1987, p. 38. Samuel 1988, p. 11. Snapper & Troyer 1989, p
Affine_transformation
Standard model in theoretical computer science
numbers), constant depth circuits, and multilinear circuits (in which every gate computes a multilinear polynomial). These restricted models have been studied
Arithmetic_circuit_complexity
Error-correcting codes used in wireless communication
based on the evaluation of multilinear polynomials with m variables and total degree at most r. Every multilinear polynomial over the finite field with
Reed–Muller_code
A cryptographic n {\displaystyle n} -multilinear map is a kind of multilinear map, that is, a function e : G 1 × ⋯ × G n → G T {\displaystyle e:G_{1}\times
Cryptographic_multilinear_map
Technique for expressing a polynomial in simpler fashion by using more variables
polynomial, polarization produces a unique symmetric multilinear form from which the original polynomial can be recovered by evaluating along a certain diagonal
Polarization of an algebraic form
Polarization_of_an_algebraic_form
In mathematics, invariant of square matrices
value of the determinant. This is a consequence of multilinearity and being alternative: by multilinearity the determinant changes by a multiple of the determinant
Determinant
Vector-valued function of multiple vectors, linear in each argument
linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function
Multilinear_map
Properties of mathematical relationships
different properties: linearity of a function (or mapping); linearity of a polynomial. An example of a linear function is the function defined by f ( x ) =
Linearity
Mathematical Theorem
Gamas's theorem is a result in multilinear algebra which states the necessary and sufficient conditions for a tensor symmetrized by an irreducible representation
Gamas's_theorem
"Smallest" commutative algebra that contains a vector space
algebra S(V) can be identified, through a canonical isomorphism, to the polynomial ring K[B], where the elements of B are considered as indeterminates. Therefore
Symmetric_algebra
French physicist and mathematician (1930–2022)
first difference vectors as the coefficients. We also call the multilinear polynomials "blossoming", following Lyle Ramshaw who in turn credited de Casteljau
Paul_de_Casteljau
1966 mathematics textbook by Serge Lang
decompositions of vector spaces under linear maps, the spectral theorem, polynomial ideals, Jordan form, convex sets and an appendix on the Iwasawa decomposition
Linear_Algebra_(book)
Topics referred to by the same term
and growth condition Multilinear form, which generalises bilinear forms to mappings VN → F Quadratic form, a homogeneous polynomial of degree two in a number
Form
Square root of the determinant of a skew-symmetric square matrix
square of a polynomial in the matrix entries, a polynomial with integer coefficients that only depends on m. When m is odd, the polynomial is zero, and
Pfaffian
group Orientation (geometry) Improper rotation Symplectic structure Multilinear algebra Tensor Classical treatment of tensors Component-free treatment
Outline_of_linear_algebra
Hypothesis in computational complexity theory
problem cannot be solved efficiently (where efficiently typically means "in polynomial time"). It is not known how to prove (unconditional) hardness for essentially
Computational hardness assumption
Computational_hardness_assumption
Theory of getting acceptably close inexact mathematical calculations
arithmetic. This is accomplished by using a polynomial of high degree, and/or narrowing the domain over which the polynomial has to approximate the function. Narrowing
Approximation_theory
States that the algebra of n by n matrices satisfies a certain identity of degree 2n
matrix rings are polynomial identity rings such that the smallest identity they satisfy has degree exactly 2n. The standard polynomial of degree n is S
Amitsur–Levitzki_theorem
Mathematical function generalizing the determinant and permanent
properties with determinant and permanent. In particular, the immanant is multilinear in the rows and columns of the matrix; and the immanant is invariant
Immanant
Homogeneous polynomial of degree 3
In mathematics, a cubic form is a homogeneous polynomial of degree 3, and a cubic hypersurface is the zero set of a cubic form. In the case of a cubic
Cubic_form
Set-to-real map with diminishing returns
such that each 0 ≤ x i ≤ 1 {\displaystyle 0\leq x_{i}\leq 1} . Then the multilinear extension is defined as F ( x ) = ∑ S ⊆ Ω f ( S ) ∏ i ∈ S x i ∏ i ∉ S
Submodular_set_function
Type of cryptographic software obfuscation
arguments Constant-round concurrent zero-knowledge protocols Multilinear maps with bounded polynomial degrees Injective trapdoor functions Fully homomorphic
Indistinguishability obfuscation
Indistinguishability_obfuscation
Concept in multilinear algebra and representation theory
In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor A {\displaystyle \mathbf
Invariants_of_tensors
Square matrices satisfy their characteristic equation
integers) satisfies its own characteristic equation. The characteristic polynomial of an n × n {\displaystyle n\times n} matrix A is defined as p A ( λ )
Cayley–Hamilton_theorem
Mathematical study of invariants under symmetries
Classically, the theory dealt with the question of explicit description of polynomial functions that do not change (are invariant) under the transformations
Invariant_theory
Polynomial with only one term
representation Monomial matrix Homogeneous polynomial Homogeneous function Multilinear form Log-log plot Power law Sparse polynomial Lang, Serge (2005). Algebra. GTM
Monomial
Coordinate-free definition of a tensor
views a tensor as an abstract object, expressing some definite type of multilinear concept. Their properties can be derived from their definitions, as linear
Tensor_(intrinsic_definition)
Concept in algebra
definition is a discriminant for a singular point on a scalar valued multilinear map. Cayley's first hyperdeterminant is defined only for hypercubes having
Hyperdeterminant
Algebraic structure with addition and multiplication
observation that one can multiply pointwise a k-multilinear form and an l-multilinear form to get a (k + l)-multilinear form. The ring structure in cohomology
Ring_(mathematics)
a polynomially reflexive space is a Banach space X, on which the space of all polynomials in each degree is a reflexive space. Given a multilinear functional
Polynomially_reflexive_space
Disproved conjecture in multilinear algebra on the rank of symmetric tensors
In mathematics, Comon's conjecture was a conjecture in multilinear algebra asserting that the rank and the symmetric rank of a symmetric tensor are always
Comon's_conjecture
Function with a multiplicative scaling behaviour
kth-degree or kth-order homogeneous function. For example, a homogeneous polynomial of degree k defines a homogeneous function of degree k. The above definition
Homogeneous_function
Mathematical operation on vector spaces
they are seen as multilinear maps (see also tensors as multilinear maps). Thus the components of the tensor product of multilinear forms can be computed
Tensor_product
Cryptographic primitives that involve lattices
homomorphic encryption, indistinguishability obfuscation, cryptographic multilinear maps, and functional encryption. Lattice problems Learning with errors
Lattice-based_cryptography
Branch of mathematical statistics
intersection of several areas of mathematics, including, for instance, multilinear algebra, commutative algebra, algebraic geometry, convex geometry, combinatorics
Algebraic_statistics
particular invariant rings, see invariants of a binary form, symmetric polynomials. For geometric terms used in invariant theory see the glossary of classical
Glossary_of_invariant_theory
Matrix whose determinant is a resultant
and Multilinear Algebra, 10 (4): 265–308, doi:10.1080/03081088108817420, ISSN 0308-1087, MR 0638124 Pan, Victor; Bini, Dario (1994). Polynomial and matrix
Bézout_matrix
Mathematical theory
viewing f as a symmetric multilinear functional on ∏ 1 k g {\textstyle \prod _{1}^{k}{\mathfrak {g}}} (see the ring of polynomial functions), let f ( Ω )
Chern–Weil_homomorphism
Decomposition in multilinear algebra
In multilinear algebra, the tensor rank decomposition or rank-R decomposition is the decomposition of a tensor as a sum of R rank-1 tensors, where R is
Tensor_rank_decomposition
lattice theory, denotes the meet or greatest lower bound operation. 3. In multilinear algebra, geometry, and multivariable calculus, denotes the exterior product
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Graphical means of performing computations in linear algebra
diagrams are a graphical means of performing computations in linear and multilinear algebra. They can be represented as (slightly modified) graphs in which
Trace_diagram
Mathematical concept in algebra
characteristic polynomial for N {\displaystyle N} is det ( x I − N ) = x n {\displaystyle \det \left(xI-N\right)=x^{n}} . The minimal polynomial for N {\displaystyle
Nilpotent_matrix
characteristic polynomial of the adjacency matrix of graphs. A similar expansion using Sachs subgraphs is also possible for permanental polynomials of graphs
Sachs_subgraph
Tensor invariant under permutations of vectors it acts on
space V is naturally isomorphic to the dual of the space of homogeneous polynomials of degree r on V. Over fields of characteristic zero, the graded vector
Symmetric_tensor
Class of mathematical software
toolbox for multilinear algebra and structured data fusion. Tensor Toolbox Multilinear algebra MATLAB software. MPCA and MPCA+LDA Multilinear subspace learning
Tensor_software
Function that is invariant under all permutations of its variables
symmetric functions are polynomial functions, which are given by the symmetric polynomials. A related notion is alternating polynomials, which change sign
Symmetric_function
Polynomial of the elements of a matrix
similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix. Both are special cases of a more general
Permanent_(mathematics)
Algorithm for reducing the dimension of tensors
properties of tensor sketches, particularly focused on applications to polynomial kernels. In this context, the sketch is required not only to preserve
Tensor_sketch
associated statistics are called U-statistics. Alternating multilinear map – Multilinear map that is 0 whenever arguments are linearly dependent Antisymmetric
Symmetrization
Mathematical formula for the number of Young tableaux
of semi-standard Young tableaux, which is a specialization of a Schur polynomial. Let λ = ( λ 1 ≥ ⋯ ≥ λ k ) {\displaystyle \lambda =(\lambda _{1}\geq \cdots
Hook_length_formula
Branch of mathematics
over a field. For more details, see Linear equation over a ring. In multilinear algebra, one considers multivariable linear transformations, that is
Linear_algebra
Mathematical proof about the permanent of matrices
completeness, and his proof of completeness of 01-permanent, both used polynomial-time Turing reductions. In this kind of reduction, a single hard instance
♯P-completeness of 01-permanent
♯P-completeness_of_01-permanent
engineering science For some history of the abstract theory see also multilinear algebra. Ricci calculus The earliest foundation of tensor theory – tensor
Glossary_of_tensor_theory
Integration for Grassmann variables
fermions. Let Λ n {\displaystyle \Lambda ^{n}} be the exterior algebra of polynomials in anticommuting elements θ 1 , … , θ n {\displaystyle \theta _{1},\dots
Berezin_integral
Sequence of operations for a task
randomized polynomial time algorithm, but not by a deterministic one: see Dyer, Martin; Frieze, Alan; Kannan, Ravi (January 1991). "A Random Polynomial-time
Algorithm
Matrix operation which flips a matrix over its diagonal
invariant factors, which implies they share the same minimal polynomial, characteristic polynomial, and eigenvalues, among other properties. A proof of this
Transpose
Numerical methods for matrix eigenvalue calculation
submatrices of normal, hermitian and symmetric matrices". Linear and Multilinear Algebra. 36 (1): 69–78. doi:10.1080/03081089308818276. Bebiano N, Furtado
Eigenvalue_algorithm
Scalar quantities representing curvature
are called n-th order differential invariants. The Riemann tensor is a multilinear operator of fourth rank acting on tangent vectors. However, it can also
Curvature_invariant
Polish mathematician
and Multilinear Algebra. 11 (2): 121–131. doi:10.1080/03081088208817437. Minc, Henryk (1987). "Theory of permanents 1982–1985". Linear and Multilinear Algebra
Henryk_Minc
techniques from integral and differential calculus as well as linear and multilinear algebra to study problems in geometry. Classically, these were problems
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Voting power index based on pivotal probability
simple games and uniform ordering probabilities. Methods based on the multilinear extension and marginal-contribution distributions provide efficient approximations
Shapley–Shubik_power_index
Subset of artificial intelligence
representation is sparse, meaning that the mathematical model has many zeros. Multilinear subspace learning algorithms aim to learn low-dimensional representations
Machine_learning
Vector satisfying some of the criteria of an eigenvector
{\displaystyle A} must be in F {\displaystyle F} . That is, the characteristic polynomial f ( x ) {\displaystyle f(x)} must factor completely into linear factors;
Generalized_eigenvector
Certain functors from the category of modules over a fixed commutative ring to itself
M} satisfying the following conditions φ {\displaystyle \varphi } is multilinear, φ {\displaystyle \varphi } is alternating in the entries indexed by
Schur_functor
American mathematician
[F69] It is a comprehensive work beginning with a detailed account of multilinear algebra and measure theory. The main body of the work is devoted to a
Herbert_Federer
over square matrices which is distributive over matrix multiplication, multilinear in the rows and columns, and takes the value of 1 {\displaystyle 1} for
Glossary_of_linear_algebra
American mathematician
differential equations". 1987: (with Jean-Lin Journé) "Polynomial growth estimates for multilinear singular integral operators", Acta Mathematica 159(1–2):
F._Michael_Christ
Mathematical identity concerning matrices
operators, polarizations, and a generalized Capelli identity", Linear & Multilinear Algebra, 10 (2): 93–102, doi:10.1080/03081088108817399 Umeda, Toru (2000)
Capelli's_identity
Sum of elements on the main diagonal
written in the notation of tensor products. Then one may consider the multilinear map V × V∗ × V × V∗ → V ⊗ V∗ given by sending (v, φ, w, ψ) to φ(w)v ⊗
Trace_(linear_algebra)
Study of discrete mathematical structures
and is closely related to q-series, special functions and orthogonal polynomials. Originally a part of number theory and analysis, partition theory is
Discrete_mathematics
Algebraic structure in linear algebra
{\displaystyle V} and W {\displaystyle W} is one of the central notions of multilinear algebra, which deals with extending notions such as linear maps to several
Vector_space
Algorithms for matrix decomposition
rational. Recently, this problem has been answered negatively. Multilinear algebra Multilinear subspace learning Tensor Tensor decomposition Tensor software
Non-negative matrix factorization
Non-negative_matrix_factorization
{\displaystyle V} is symmetric in its arguments; V {\displaystyle V} is multilinear: V ( λ K + λ ′ K ′ , K 2 , … , K n ) = λ V ( K , K 2 , … , K n ) + λ
Mixed_volume
Set of vectors used to define coordinates
set of polynomials such that there is exactly one polynomial of each degree (such as the Bernstein basis polynomials or Chebyshev polynomials) is also
Basis_(linear_algebra)
Form of encryption that allows computation on ciphertexts
(2016). "An algorithm for NTRU problems and cryptanalysis of the GGH multilinear map without a low-level encoding of zero". LMS Journal of Computation
Homomorphic_encryption
Holomorphic functions in infinite dimensions
{\widehat {D}}^{n}f(x)(y)} is the homogeneous polynomial of degree n in y associated with the multilinear operator Dnf(x). The convergence of this series
Infinite-dimensional holomorphy
Infinite-dimensional_holomorphy
Universal construction in multilinear algebra
before. Braided vector space Braided Hopf algebra Monoidal category Multilinear algebra Fock space Bourbaki, Nicolas (1989). Algebra I. Chapters 1-3
Tensor_algebra
Expression that may be integrated over a region
differential form Equivariant differential form Calculus on Manifolds Multilinear form Polynomial differential form Presymplectic form Cartan, Élie (1899), "Sur
Differential_form
Israeli-American mathematician (born 1944)
doi:10.1103/PhysRevA.22.618 "Convex spectral functions", Linear and Multilinear Algebra, vol. 9, no. 4, 1981, 299–316. doi:10.1080/03081088108817381
Shmuel_Friedland
theorem (geometry) Exchange theorem (linear algebra) Gamas's Theorem (multilinear algebra) Gershgorin circle theorem (matrix theory) Inverse eigenvalues
List_of_theorems
Array of numbers
the eigenvalues of a square matrix are the roots of its characteristic polynomial, det ( λ I − A ) {\displaystyle \det(\lambda I-A)} . Matrix theory is
Matrix_(mathematics)
Algebra based on a vector space with a quadratic form
• Principal ideal domain • Euclidean domain • Field • Finite field • Polynomial ring • Formal power series ring Algebraic number theory • Algebraic number
Clifford_algebra
Classification scheme for mathematics
Field theory and polynomials 13: Commutative algebra (Commutative rings and algebras) 14: Algebraic geometry 15: Linear and multilinear algebra; matrix
Mathematics Subject Classification
Mathematics_Subject_Classification
Iranian-American mathematician
ISBN 978-3-031-40002-5. Kahrobaei, D.; Stanojkovski, M. (2023). "Cryptographic multilinear maps using pro-p groups". Advances in Mathematics of Communications.
Delaram_Kahrobaei
Mathematical operation on matrices
Shayle R. (1983). "On the history of the kronecker product". Linear and Multilinear Algebra. 14 (2): 113–120. doi:10.1080/03081088308817548. hdl:1813/32834
Kronecker_product
Ring produced from two fields
{\displaystyle ac\otimes bd} (see Tensor product of algebras). This formula is multilinear over N in each variable; and so defines a ring structure on the tensor
Tensor_product_of_fields
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