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In algebra, a central polynomial for n-by-n matrices is a polynomial in non-commuting variables that is non-constant but yields a scalar matrix whenever
Central_polynomial
Polynomials arising in knot theory
theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant
HOMFLY_polynomial
Type of mathematical expression
functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic
Polynomial
Estimate of time taken for running an algorithm
Problems for which a deterministic polynomial-time algorithm exists belong to the complexity class P, which is central in the field of computational complexity
Time_complexity
System of complete and orthogonal polynomials
mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of
Legendre_polynomials
Error-detecting code for detecting data changes
systems get a short check value attached, based on the remainder of a polynomial division of their contents. On retrieval, the calculation is repeated
Cyclic_redundancy_check
Algebraic encoding of graph connectivity
The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays
Tutte_polynomial
historically important, finding the roots of higher degree polynomials no longer play a central role in mathematics and computational mathematics, with one
Polynomial_root-finding
Statistics concept
In statistics, polynomial regression is a form of regression analysis in which the relationship between the independent variable x and the dependent variable
Polynomial_regression
Orthogonal symmetric polynomial family
In mathematics, Macdonald polynomials Pλ(x; t,q) are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987
Macdonald_polynomials
Complexity class
Each input to the problem is associated with a collection of short (polynomial length) solutions, which might or might not validly solve the input. The
NP-completeness
Algorithm to smooth data points
fitting successive sub-sets of adjacent data points with a low-degree polynomial by the method of linear least squares. When the data points are equally
Savitzky–Golay_filter
Polynomial whose coefficients are all 1 or −1
In mathematics, a Littlewood polynomial is a polynomial whose coefficients are all either 1 {\displaystyle 1} or − 1 {\displaystyle -1} . Equivalently
Littlewood_polynomial
In ring theory, a branch of mathematics, a ring R is a polynomial identity ring if there is, for some N > 0, an element P ≠ 0 of the free algebra, Z⟨X1
Polynomial_identity_ring
Generating polynomial of the number of ways to place non-attacking rooks on a chessboard
In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board that looks like
Rook_polynomial
Complexity class
every problem L which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from L to H. That is, assuming a solution
NP-hardness
Property whereby no efficient algorithm can distinguish two distributions
computationally indistinguishable if for any non-uniform probabilistic polynomial time algorithm A, the following quantity is a negligible function in n:
Computational indistinguishability
Computational_indistinguishability
Theorem in geometric group theory
is then the least degree of any such polynomial function p. A nilpotent group G is a group with a lower central series terminating in the identity subgroup
Gromov's theorem on groups of polynomial growth
Gromov's_theorem_on_groups_of_polynomial_growth
a polynomial ring in one variable. For example, a central polynomial is an element of the ring F n {\displaystyle F_{n}} that will map to a central element
Generic_matrix_ring
Canonical solutions of the general Legendre equation
In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre differential equation ( 1 − x 2 ) d 2 d x 2 P
Associated Legendre polynomials
Associated_Legendre_polynomials
Function in algebraic graph theory
The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a
Chromatic_polynomial
Discrete analog of a derivative
the polynomial is 36x. Subtracting out the third term: Without any pairwise differences, it is found that the 4th and final term of the polynomial is the
Finite_difference
Visualization of the prime numbers
spiral correspond to quadratic polynomials, and certain such polynomials, such as Euler's prime-generating polynomial x2 − x + 41, are believed to produce
Ulam_spiral
Statistical approach
more complicated design, such as a central composite design can be implemented to estimate a second-degree polynomial model, which is still only an approximation
Response_surface_methodology
Mathematics concept
In mathematics, the Romanovski polynomials are one of three finite subsets of real orthogonal polynomials discovered by Vsevolod Romanovsky (Romanovski
Romanovski_polynomials
Approximation of a function by a polynomial
by a polynomial of degree k {\textstyle k} , called the k {\textstyle k} -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the
Taylor's_theorem
Makhzen. Exact value is an irrational number which is a root of a quartic polynomial (sequence A230582 in the OEIS). See Flag of Nepal § Aspect ratio. See
List of national flags of sovereign states
List_of_national_flags_of_sovereign_states
considered to be "between" the central simple algebra version and the Artinian ring version. This is because simple polynomial identity rings are Artinian
Double_centralizer_theorem
Process of constructing a curve that has the best fit to a series of data points
for higher order polynomial equations. If there are more than n + 1 constraints (n being the degree of the polynomial), the polynomial curve can still
Curve_fitting
Expression for sums of powers
{\displaystyle \sum _{k=1}^{n}k^{p}=1^{p}+2^{p}+3^{p}+\cdots +n^{p}} as a polynomial in n {\displaystyle n} . In modern notation, Faulhaber's formula is ∑
Faulhaber's_formula
18 mathematical problems stated in 1998
"A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time". Foundations of Computational Mathematics. to
Smale's_problems
Mathematical connection between field theory and group theory
introduced the subject for studying roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms
Galois_theory
Type of polynomial sequence
Abel polynomials The Bernoulli polynomials The Euler polynomials The central factorial polynomials The Hermite polynomials The Laguerre polynomials The
Sheffer_sequence
Israeli mathematician (1921–1994)
central polynomials for matrix algebras, and the study of sequences of codimensions and cocharacters of PI-algebras. His work on central polynomials simplified
Shimshon_Amitsur
central role in the study of counting points on elliptic curves in Schoof's algorithm. The set of division polynomials is a sequence of polynomials in
Division_polynomials
Subfield of computational complexity theory
algorithms, but in the next decade they became central to lower bounds for problems that already run in polynomial time. In 2004, Ryan Williams gave the reduction
Fine_grained_complexity
American mathematician and chess player
the central polynomials, which have applications to polynomial identity rings. With Vesselin Drensky, Formanek is the author of the book Polynomial Identity
Edward_W._Formanek
Mathematical functions
Mittag-Leffler polynomials are the polynomials gn(x) or Mn(x) studied by Mittag-Leffler (1891). Mn(x) is a special case of the Meixner polynomial Mn(x;b,c)
Mittag-Leffler_polynomials
Polynomial sequence
The Bernoulli polynomials of the second kind ψn(x), also known as the Fontana–Bessel polynomials, are the polynomials defined by the following generating
Bernoulli polynomials of the second kind
Bernoulli_polynomials_of_the_second_kind
Abstraction of linear independence of vectors
isomorphic matroids have the same polynomial. The characteristic polynomial of M – sometimes called the chromatic polynomial, although it does not count colorings
Matroid
extension are called Ore polynomials. Ore extensions appear in several natural contexts, including skew and differential polynomial rings, group algebras
Ore_extension
can be solved in quasi-polynomial time in the combined size of its input and output, but whether they can be solved in polynomial time is an open problem
Monotone_dualization
American mathematician
of the bracket polynomial and the Kauffman polynomial. Kauffman was valedictorian of his graduating class at Norwood Norfolk Central High School in 1962
Louis_Kauffman
Computation of a cyclic redundancy check is derived from the mathematics of polynomial division, modulo two. In practice, it resembles long division of the binary
Computation of cyclic redundancy checks
Computation_of_cyclic_redundancy_checks
Algorithms for zeros of functions
historically important, finding the roots of higher degree polynomials no longer play a central role in mathematics and computational mathematics, with one
Root-finding_algorithm
Concepts from linear algebra
the roots of a polynomial with degree 5 or more. (Generality matters because any polynomial with degree n is the characteristic polynomial of some companion
Eigenvalues_and_eigenvectors
Methods of error detection and correction in communications
after division in the ring of polynomials over GF(2) (the finite field of integers modulo 2). That is, the set of polynomials where each coefficient is either
Mathematics of cyclic redundancy checks
Mathematics_of_cyclic_redundancy_checks
Study of abstract algebraic structures
central interest are the free commutative algebras, namely the polynomial algebras. In this particularly simple and important case of the polynomial algebra
Algebra_representation
Branch of mathematics
above example). Polynomials of degree one are called linear polynomials. Linear algebra studies systems of linear polynomials. A polynomial is said to be
Algebra
Mathematical construct in computer algebra
Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring K [ x 1 , … , x n ] {\displaystyle K[x_{1},\ldots ,x_{n}]} over a
Gröbner_basis
shown in the following figures. If the original density is a piecewise polynomial, as it is in the example, then so are the sum densities, of increasingly
Illustration of the central limit theorem
Illustration_of_the_central_limit_theorem
Number divisible only by 1 and itself
and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available
Prime_number
polytope associated with a multivariate polynomial that can be used in the asymptotic analysis of those polynomials. It is a generalization of the Kruskal–Newton
Newton_polytope
On the approximate structure of sets whose sumset is small
from a new proof by Imre Z. Ruzsa (1992,1994). Mei-Chu Chang proved new polynomial estimates for the size of arithmetic progressions arising in the theorem
Freiman's_theorem
Coefficient used in numerical approximation
basically computed by fitting and deriving a 2 p {\displaystyle 2p} -th order polynomial to a window of 2 p + 1 {\displaystyle 2p+1} points. Consequently, the
Finite_difference_coefficient
Algebraic structure with addition and multiplication
complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. More formally, a ring
Ring_(mathematics)
Branch of mathematics
geometrical problems. Classically, it studies zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The
Algebraic_geometry
Polynomial equation of degree two
non-negative integers, and therefore it is a polynomial equation. In particular, it is a second-degree polynomial equation, since the greatest power is two
Quadratic_equation
Mathematics award
Neumann algebras and geometric topology. As a result, he found a new polynomial invariant for knots and links in 3-space." Shigefumi Mori Kyoto University
Fields_Medal
Sum of elements on the main diagonal
the characteristic polynomial, possibly changed of sign, according to the convention in the definition of the characteristic polynomial. If a is a square
Trace_(linear_algebra)
Egyptian mathematician
27, 1944, in Cairo, Egypt) is a mathematician working on orthogonal polynomials and special functions. Ismail received his bachelor's degree from Cairo
Mourad_Ismail
Type of two-dimensional barcode
. The primitive polynomial is x 8 + x 4 + x 3 + x 2 + 1 {\displaystyle x^{8}+x^{4}+x^{3}+x^{2}+1} , corresponding to the polynomial number 285, with
QR_code
Polynomial-time algorithm for the assignment problem
combinatorial optimization algorithm that solves the assignment problem in polynomial time and which anticipated later primal–dual methods. It was developed
Hungarian_algorithm
Algebraic structure in linear algebra
all polynomials p ( t ) {\displaystyle p(t)} forms an algebra known as the polynomial ring: using that the sum of two polynomials is a polynomial, they
Vector_space
Mathematical tool for studying knots
answer the question is using knot polynomials, which are invariants of the knot. If two diagrams have different polynomials, they represent different knots
Skein_relation
Conjecture on zeros of the zeta function
Riemann hypothesis implies that one can test if a number is prime in polynomial time via the Miller test. In 2002, Manindra Agrawal, Neeraj Kayal and
Riemann_hypothesis
Special mathematical functions defined on the surface of a sphere
radial dependence r ℓ {\displaystyle r^{\ell }} from the above-mentioned polynomial of degree ℓ {\displaystyle \ell } ; the remaining factor can be regarded
Spherical_harmonics
Skeletonized version of algebraic geometry
In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication
Tropical_geometry
Construction of a larger algebraic field by "adding elements" to a smaller field
extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry
Field_extension
Unicode denominator & numerator glyphs
characters including a full set of Arabic numerals. These characters allow any polynomial, chemical and certain other equations to be represented in plain text
Unicode subscripts and superscripts
Unicode_subscripts_and_superscripts
Family of polynomials
coefficients (also called Gaussian coefficients, Gaussian numbers, Gaussian polynomials, or q-binomial coefficients) are q-analogs of the binomial coefficients
Gaussian_binomial_coefficient
Degree of connectedness within a graph
in polynomial time, it is NP-hard to find sets of a given size k {\displaystyle k} maximizing group centrality for many measures. Alpha centrality Group
Centrality
Isomorphism of commutative rings constructed in the theory of Lie algebras
element acting on polynomials by h ↦ − h {\displaystyle h\mapsto -h} . The subalgebra of Weyl-invariant polynomials in the full polynomial algebra K [ h ]
Harish-Chandra_isomorphism
Polynomial with all terms of degree two
mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, 4 x 2 + 2 x y
Quadratic_form
Class in computational complexity theory
problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors. In other words, a problem with input size n is in
NC_(complexity)
Topological quantum field theory
calculate knot invariants and three-manifold invariants such as the Jones polynomial. Particularly, Chern–Simons theory is specified by a choice of simple
Chern–Simons_theory
System where changes of output are not proportional to changes of input
equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words
Nonlinear_system
In mathematics, element that equals its square
idempotent f. For example, this could be applied to x ∈ Z[x], or any polynomial f ∈ k[x1, ..., xn]. There is a circle of idempotents in the ring of split-quaternions
Idempotent_(ring_theory)
Problem of determining if a Boolean formula could be made true
solves each SAT problem (where "efficiently" means "deterministically in polynomial time"). Although such an algorithm is generally believed not to exist
Boolean satisfiability problem
Boolean_satisfiability_problem
law – Jacques Charles Chebyshev distance, equation, filter, linkage, polynomials – Pafnuty Chebyshev Chebyshev's inequality (a.k.a. Bienaymé–Chebyshev
Scientific phenomena named after people
Scientific_phenomena_named_after_people
Complete subgraph added to a random graph
clique problem; it may be solved in quasi-polynomial time but is conjectured not to be solvable in polynomial time for intermediate values of the clique
Planted_clique
Tabular arrangement of the chemical elements
equation for this potential can be described analytically with Gegenbauer polynomials. As v {\displaystyle v} passes through each of these values, a manifold
Periodic_table
English polymath (1642–1727)
Newton's method, the Newton polygon, and classified cubic plane curves (polynomials of degree three in two variables). Newton is also a founder of the theory
Isaac_Newton
Approximation method in statistics
a linear one, and thus the core calculation is similar in both cases. Polynomial least squares describes the variance in a prediction of the dependent
Least_squares
Inherent difficulty of computational problems
T ( n ) {\displaystyle T(n)} is a polynomial in n {\displaystyle n} , then the algorithm is said to be a polynomial time algorithm. Cobham's thesis argues
Computational complexity theory
Computational_complexity_theory
variety Given r in R, the polynomial ∏ g ∈ G ( t − g ⋅ r ) {\displaystyle \prod _{g\in G}(t-g\cdot r)} is a monic polynomial over RG and has r as one of
Fixed-point_subring
Generalization of perpendicularity
fact is a central one in Fourier series. Various polynomial sequences named for mathematicians of the past are sequences of orthogonal polynomials. In particular:
Orthogonality_(mathematics)
(1992), and Jones’s cons-free programming language characterisation of polynomial time (1999). ICC is also concerned with the practical realization of functional
Implicit computational complexity
Implicit_computational_complexity
Polynomial equation whose integer solutions are sought
In mathematics, a Diophantine equation is a polynomial equation with integer coefficients, for which only integer solutions are of interest. A linear Diophantine
Diophantine_equation
Fractal named after mathematician Benoit Mandelbrot
parameters c {\displaystyle c} for which the Julia set of the corresponding polynomial forms a connected set. In the same way, the boundary of the Mandelbrot
Mandelbrot_set
Mathematical element
said to be integral over a subring A of B if b is a root of some monic polynomial over A. If A, B are fields, then the notions of "integral over" and of
Integral_element
English mathematician, philosopher, and engineer (1791–1871)
with what he called the difference engine, made to compute values of polynomial functions. It was created to calculate a series of values automatically
Charles_Babbage
Pseudocylindrical compromise map projection
the creation of new projections. Subsequently, Bojan Šavrič developed a polynomial expression of the projection. The projection may also be referred to as
Natural_Earth_projection
Set with associative invertible operation
way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both
Group_(mathematics)
Analog multiplexing technique used in early telephone systems
Darlington, Sidney (April 1952). "Network Synthesis Using Tchebycheff Polynomial Series†". Bell System Technical Journal. 31 (4): 613–665. Bibcode:1952BSTJ
Carrier_telephony
Class of nonparametric methods
popular embedding kernels k {\displaystyle k} (e.g. the Gaussian kernel or polynomial kernel), or can be accurately empirically estimated from i.i.d. samples
Kernel embedding of distributions
Kernel_embedding_of_distributions
Private university in Pasadena, California
investigations of polynomials. Narendra Karmarkar (MS 1979) is known for the interior point method, a polynomial algorithm for linear programming
California Institute of Technology
California_Institute_of_Technology
Method to solve optimization problems
polynomial-time algorithm? Does LP admit a strongly polynomial-time algorithm to find a strictly complementary solution? Does LP admit a polynomial-time
Linear_programming
Model that describes the programmable interface of a computer processor
than 3 operands (registers or memory accesses), such as the VAX "POLY" polynomial evaluation instruction. Due to the large number of bits needed to encode
Instruction_set_architecture
Topics referred to by the same term
1946 Hungarians/Magyars, ethnic groups in Hungary Hungarian algorithm, a polynomial time algorithm for solving the assignment problem Hungarian language,
Hungarian
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