DOSEN PROFIL LENGKAP

In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the...

Quasi-polynomial time algorithms are algorithms whose running time exhibits quasi-polynomial growth, a type of behavior that may be slower than polynomial time...

In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues...

the solution to a problem can be checked in polynomial time, must the problem be solvable in polynomial time? More unsolved problems in computer science...

of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function...

computer science In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems. NP is...

In statistics, polynomial regression is a form of regression analysis in which the relationship between the independent variable x and the dependent variable...

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as T n ( x ) {\displaystyle T_{n}(x)}...

In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a...

Newton polynomial, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points. The Newton polynomial is sometimes...

mathematics, a polynomial lemniscate or polynomial level curve is a plane algebraic curve of degree 2n, constructed from a polynomial p with complex coefficients...

especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally...

systems get a short check value attached, based on the remainder of a polynomial division of their contents. On retrieval, the calculation is repeated...

In computational complexity theory, a polynomial-time reduction is a method for solving one problem using another. One shows that if a hypothetical subroutine...

A polynomial P is annihilating or called an annihilating polynomial in linear algebra and operator theory if the polynomial considered as a function of...

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...

mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of...

Important graph polynomials include: The characteristic polynomial, based on the graph's adjacency matrix. The chromatic polynomial, a polynomial whose values...

In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the...

mathematics, the order of a polynomial may refer to: the degree of a polynomial, that is, the largest exponent (for a univariate polynomial) or the largest sum...

In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through...

precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number...

computer science, a polynomial-time algorithm is – generally speaking – an algorithm whose running time is upper-bounded by some polynomial function of the...

In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The...

elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed...

function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields...

polynomial functor is an endofunctor on the category V {\displaystyle {\mathcal {V}}} of finite-dimensional vector spaces that depends polynomially on...

subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n...

{\displaystyle n} -th cyclotomic polynomial, for any positive integer n {\displaystyle n} , is the unique irreducible polynomial with integer coefficients that...

GCD or gcd) of two polynomials is a polynomial, of the highest possible degree, which is a factor of both the two original polynomials. This concept is...

In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version...

science, a function f ( n ) {\displaystyle f(n)} is said to exhibit quasi-polynomial growth when it has an upper bound of the form f ( n ) = 2 O ( ( log ⁡...

an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of...

In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander...

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...

numerical analysis, a Bernstein polynomial is a polynomial expressed as a linear combination of Bernstein basis polynomials. The idea is named after mathematician...

In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: x y ″...

variables, polynomial convexity is a notion of convexity for compact subsets of complex Euclidean space defined using complex polynomials. It is analogous...

mathematics, a polynomial decomposition expresses a polynomial f as the functional composition g ∘ h {\displaystyle g\circ h} of polynomials g and h, where...

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...

theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant...

In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that...

symmetric polynomial is a polynomial P(X1, X2, ..., Xn) in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally...

Polynomial chaos (PC), also called polynomial chaos expansion (PCE) and Wiener chaos expansion, is a method for representing a random variable in terms...

Finding the roots of polynomials is a long-standing problem that has been extensively studied throughout the history and substantially influenced the...

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...

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...

root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number...

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: signal processing as Hermitian wavelets...

the polynomial method is an algebraic approach to combinatorics problems that involves capturing some combinatorial structure using polynomials and proceeding...

mathematics, Conway polynomial can refer to: the Alexander–Conway polynomial in knot theory the Conway polynomial (finite fields) the polynomial of degree 71...

non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since...

In mathematics, a positive polynomial (respectively non-negative polynomial) on a particular set is a polynomial whose values are positive (respectively...

an algebraic equation or polynomial equation is an equation of the form P = 0 {\displaystyle P=0} , where P is a polynomial, usually with rational numbers...

In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example...

In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P n ( α , β ) ( x ) {\displaystyle P_{n}^{(\alpha ,\beta )}(x)} are...

In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables, indexed by partitions, that generalize the...

machine in polynomial time, with an error probability of less than 1/2 for all instances. The abbreviation PP refers to probabilistic polynomial time. The...

In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a...

In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant...

the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) is an application of Euclidean division of polynomials. It states...

In coding theory, a polynomial code is a type of linear code whose set of valid code words consists of those polynomials (usually of some fixed length)...

Polynomial Matrix Spectral Factorization or Matrix Fejer–Riesz Theorem is a tool used to study the matrix decomposition of polynomial matrices. Polynomial...

from an arbitrary field, its reciprocal polynomial or reflected polynomial, denoted by p∗ or pR, is the polynomial p ∗ ( x ) = a n + a n − 1 x + ⋯ + a 0...

Polynomial identity may refer to: Algebraic identities of polynomials (see Factorization) Polynomial identity ring Polynomial identity testing This disambiguation...

orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The...

In algebra, a multivariate polynomial f ( x ) = ∑ α a α x α , where  α = ( i 1 , … , i r ) ∈ N r , and  x α = x 1 i 1 ⋯ x r i r , {\displaystyle f(x)=\sum...

In algebra, an alternating polynomial is a polynomial f ( x 1 , … , x n ) {\displaystyle f(x_{1},\dots ,x_{n})} such that if one switches any two of the...

mathematics, the minimal polynomial of an element α {\displaystyle \alpha } of an extension field of a field is, roughly speaking, the polynomial of lowest degree...

In mathematics, a permutation polynomial (for a given ring) is a polynomial that acts as a permutation of the elements of the ring, i.e. the map x ↦ g...

mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the...

In computer science (particularly algorithmics), a polynomial-time approximation scheme (PTAS) is a type of approximation algorithm for optimization problems...

In mathematics, exponential polynomials are functions on fields, rings, or abelian groups that take the form of polynomials in a variable and an exponential...

In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(α) n(x) are orthogonal polynomials on the interval [−1,1] with respect to the weight...

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...

bounded-error probabilistic polynomial time (BPP) is the class of decision problems solvable by a probabilistic Turing machine in polynomial time with an error...

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...

introduced the subject for studying roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms...

In mathematics, a generic polynomial refers usually to a polynomial whose coefficients are indeterminates. For example, if a, b, and c are indeterminates...

In mathematics, Schubert polynomials are generalizations of Schur polynomials that represent cohomology classes of Schubert cycles in flag varieties. They...

specifically that k {\displaystyle k} points on the polynomial uniquely determines a polynomial of degree less than or equal to k − 1 {\displaystyle...

theory, a numeric algorithm runs in pseudo-polynomial time if its running time is bounded from above by a polynomial function of the two variables: the numeric...

and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several...

structures) is said to have polynomial delay if the time between the output of any one structure and the next is bounded by a polynomial function of the input...

algebra, a polynomial map or polynomial mapping P : V → W {\displaystyle P:V\to W} between vector spaces over an infinite field k is a polynomial in linear...

Minimal polynomial can mean: Minimal polynomial (field theory) Minimal polynomial (linear algebra) This disambiguation page lists mathematics articles...

Gromov's theorem on groups of polynomial growth, first proved by Mikhail Gromov, characterizes finitely generated groups of polynomial growth, as those groups...

In mathematics, a Laurent polynomial (named after Pierre Alphonse Laurent) in one variable over a field F {\displaystyle \mathbb {F} } is a linear combination...

oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation...

computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. It is named after William George Horner, although it is much...

Gauss's lemma, named after Carl Friedrich Gauss, is a theorem about polynomials over the integers, or, more generally, over a unique factorization domain...

In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series...

In mathematics, Meixner polynomials (also called discrete Laguerre polynomials) are a family of discrete orthogonal polynomials introduced by Josef Meixner (1934)...

degree bound n {\displaystyle n} , a minimax polynomial approximation algorithm will find a polynomial p {\displaystyle p} of degree at most n {\displaystyle...

In complexity theory, ZPP (zero-error probabilistic polynomial time) is the complexity class of problems for which a probabilistic Turing machine exists...

In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number...

theory, bounded-error quantum polynomial time (BQP) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability...

In mathematics, the Hamiltonian cycle polynomial of an n×n-matrix is a polynomial in its entries, defined as ham ⁡ ( A ) = ∑ σ ∈ H n ∏ i = 1 n a i , σ...

of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one variable...

titled "Polynomial Codes over Certain Finite Fields". The original encoding scheme described in the Reed and Solomon article used a variable polynomial based...

In algebra, the Vandermonde polynomial of an ordered set of n variables X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} , named after Alexandre-Théophile...

mathematics, a polynomial matrix or matrix of polynomials is a matrix whose elements are univariate or multivariate polynomials. Equivalently, a polynomial matrix...

mathematics, a polynomial Diophantine equation is an indeterminate polynomial equation for which one seeks solutions restricted to be polynomials in the indeterminate...

In combinatorial mathematics, the necklace polynomial, or Moreau's necklace-counting function, introduced by C. Moreau (1872), counts the number of distinct...

algorithms, an algorithm is said to take quasi-polynomial time if its time complexity is quasi-polynomially bounded. That is, there should exist a constant...

Minimum polynomial can refer to: Minimal polynomial (field theory) Minimal polynomial (linear algebra) This disambiguation page lists articles associated...

spiral correspond to quadratic polynomials, and certain such polynomials, such as Euler's prime-generating polynomial x2 − x + 41, are believed to produce...

mathematics, a matrix polynomial is a polynomial with square matrices as variables. Given an ordinary, scalar-valued polynomial P ( x ) = ∑ i = 0 n a...

mathematical field of knot theory, the bracket polynomial (also known as the Kauffman bracket) is a polynomial invariant of framed links. Although it is not...

paths of a polynomial-time non-deterministic Turing machine. The problem is #P-hard, meaning that every other problem in #P has a polynomial-time Turing...

All-one polynomials Abel polynomials Bell polynomials Bernoulli polynomials Cyclotomic polynomials Dickson polynomials Fibonacci polynomials Lagrange...

In mathematics and computer science, polynomial evaluation refers to computation of the value of a polynomial when its indeterminates are substituted for...

resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root...

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...

In machine learning, the polynomial kernel is a kernel function commonly used with support vector machines (SVMs) and other kernelized models, that represents...

This is a list of polynomial topics, by Wikipedia page. See also trigonometric polynomial, list of algebraic geometry topics. Degree: The maximum exponents...

In computational complexity theory, exact quantum polynomial time (EQP or sometimes QP) is the class of decision problems that can be solved by a quantum...

difference engine is an automatic mechanical calculator designed to tabulate polynomial functions. It was designed in the 1820s, and was created by Charles Babbage...

difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selberg's polynomials, and...

Rational Polynomial Coefficients (RPCs) provide a compact representation of a ground-to-image geometry, allowing photogrammetric processing without requiring...

In mathematics, a Hurwitz polynomial (named after German mathematician Adolf Hurwitz) is a polynomial whose roots (zeros) are located in the left half-plane...

mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers...

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...

amount of memory that is polynomial in the input length (polynomial space) and if every other problem that can be solved in polynomial space can be transformed...

orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as...

solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. Cobham's thesis holds that P is the class of...

arithmetic as a polynomial mod 2. This means that the coefficients of the polynomial must be 1s or 0s. This is called the feedback polynomial or reciprocal...

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...

the Kostant polynomials, named after Bertram Kostant, provide an explicit basis of the ring of polynomials over the ring of polynomials invariant under...

the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in...

factorization be done in polynomial time on a classical (non-quantum) computer? Can the discrete logarithm be computed in polynomial time on a classical (non-quantum)...

actual function as possible. This is typically done with polynomial or rational (ratio of polynomials) approximations. The objective is to make the approximation...

is a method of polynomial interpolation, which generalizes Lagrange interpolation. Lagrange interpolation allows computing a polynomial of degree less...

In algebra, a multilinear polynomial is a multivariate polynomial that is linear (meaning affine) in each of its variables separately, but not necessarily...

process modeling), polynomial functions and rational functions are sometimes used as an empirical technique for curve fitting. A polynomial function is one...

this interpolant with a polynomial of higher degree. Consider again the problem given above. The following sixth degree polynomial goes through all the seven...

coefficients. The primitive part of such a polynomial is the quotient of the polynomial by its content. Thus a polynomial is the product of its primitive part...

In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions. The...

Polynomial texture mapping (PTM), also known as Reflectance Transformation Imaging (RTI), is a technique of imaging and interactively displaying objects...

primitive polynomial may refer to: Primitive polynomial (field theory), a minimal polynomial of an extension of finite fields Primitive polynomial (ring theory)...

fields of graph theory and combinatorics, a matching polynomial (sometimes called an acyclic polynomial) is a generating function of the numbers of matchings...

linear algebra, the minimal polynomial μA of an n × n {\displaystyle n\times n} matrix A over a field F is the monic polynomial μA over F of least degree...

In mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct...

example, 3 × 5 is an integer factorization of 15, and (x − 2)(x + 2) is a polynomial factorization of x2 − 4. Factorization is not usually considered meaningful...

mathematics, a field F is algebraically closed if every non-constant polynomial with coefficients in F has a root in F. In other words, a field is algebraically...

is a special type of piecewise polynomial called a spline. That is, instead of fitting a single, high-degree polynomial to all of the values at once, spline...

Wilkinson's polynomial is a specific polynomial which was used by James H. Wilkinson in 1963 to illustrate a difficulty when finding the roots of a polynomial: the...

In mathematics, the Dickson polynomials, denoted Dn(x,α), form a polynomial sequence introduced by L. E. Dickson (1897). They were rediscovered by Brewer...

equation is a polynomial equation (commonly called also an algebraic equation) in which the two sides are polynomials. The sides of a polynomial equation contain...

of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations f1 = 0, ..., fh = 0 where the fi are polynomials in...

order polynomial is a polynomial studied in mathematics, in particular in algebraic graph theory and algebraic combinatorics. The order polynomial counts...

In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (1540-1603)...

geometrical problems. Classically, it studies zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The...

mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field GF(pm). This means that a polynomial F(X) of degree m...

known for the introduction and development of the bracket polynomial and the Kauffman polynomial. Kauffman was valedictorian of his graduating class at Norwood...

In mathematics, a square-free polynomial is a univariate polynomial (over a field or an integral domain) that has no multiple root in an algebraically...

In mathematics, polynomial identity testing (PIT) is the problem of efficiently determining whether two multivariate polynomials are identical. More formally...

In mathematics, a univariate polynomial of degree n with real or complex coefficients has n complex roots (if counted with their multiplicities). They...

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]...

In mathematics, the Bernstein–Sato polynomial is a polynomial related to differential operators, introduced independently by Joseph Bernstein (1971) and...

isomorphic matroids have the same polynomial. The characteristic polynomial of M – sometimes called the chromatic polynomial, although it does not count colorings...

desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem...

it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion...

zonal polynomial is a multivariate symmetric homogeneous polynomial. The zonal polynomials form a basis of the space of symmetric polynomials. Zonal...

a binomial is a polynomial that is the sum of two terms, each of which is a monomial. It is the simplest kind of a sparse polynomial after the monomials...

described by René Descartes in his La Géométrie, counts the roots of a polynomial by examining sign changes in its coefficients. The number of positive...

In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition...

polynomial is a polynomial that is orthogonal to several different measures. Biorthogonal polynomials are a generalization of orthogonal polynomials and...

In mathematics, an invariant polynomial is a polynomial P {\displaystyle P} that is invariant under a group Γ {\displaystyle \Gamma } acting on a vector...

analyzed using linear methods. Polynomial signal processing is a type of non-linear signal processing, where polynomial systems may be interpreted as conceptually...

In mathematics, Macdonald polynomials Pλ(x; t,q) are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987...

by a polynomial in the length of the input. A problem is said to be strongly NP-hard if a strongly NP-complete problem has a pseudo-polynomial reduction...

regression or local polynomial regression, also known as moving regression, is a generalization of the moving average and polynomial regression. Its most...

set of basic functions depends on the context. For example, if one adds polynomial roots to the basic functions, the functions that have a closed form are...

probabilistic polynomial identity testing. Identity testing is the problem of determining whether a given multivariate polynomial is the 0-polynomial, the polynomial...

mathematics, a quasi-polynomial (sometimes called pseudo-polynomial) is a generalization of polynomials. While the coefficients of a polynomial come from a ring...

the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties...

ANF are also known as ring sum normal form (RSNF or RNF), Zhegalkin polynomials (Russian: полиномы Жегалкина), or Positive Polarity (or parity) Reed–Muller...

of irreducible monic polynomials. There are efficient algorithms for testing polynomial irreducibility and factoring polynomials over finite fields. They...

theory, a Kazhdan–Lusztig polynomial P y , w ( q ) {\displaystyle P_{y,w}(q)} is a member of a family of integral polynomials introduced by David Kazhdan...

In mathematics, a form (i.e. a homogeneous polynomial) h(x) of degree 2m in the real n-dimensional vector x is sum of squares of forms (SOS) if and only...

Littlewood polynomial is a polynomial all of whose coefficients are +1 or −1. Littlewood's problem asks for bounds on the values of such a polynomial on the...

mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients...

generated homology, the Poincaré polynomial is defined as the generating function of its Betti numbers, via the polynomial where the coefficient of x n {\displaystyle...

In the context of the characteristic polynomial of a differential equation or difference equation, a polynomial is said to be stable if either: all its...

cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form, that is, by its values and first derivatives...

This method can be generalized to give the roots of cubic polynomials and quartic polynomials, and leads to Galois theory, which allows one to understand...

if and only if for every no-instance we have a polynomial-length "certificate" and there is a polynomial-time algorithm that can be used to verify any...

In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling...

theory. A knot polynomial is a knot invariant that is a polynomial. Well-known examples include the Jones polynomial, the Alexander polynomial, and the Kauffman...

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...

Touchard polynomials, studied by Jacques Touchard (1956), also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence...

Kravchuk polynomials or Krawtchouk polynomials (also written using several other transliterations of the Ukrainian surname Кравчу́к) are discrete orthogonal...

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...

In computational complexity theory, polynomial creativity is a theory analogous to the theory of creative sets in recursion theory and mathematical logic...

Polynomial conjoint measurement is an extension of the theory of conjoint measurement to three or more attributes. It was initially developed by the mathematical...

usual multiplication of polynomials, but with coefficients multiplied modulo p and polynomials multiplied modulo the polynomial m(x). This representation...

In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike...

Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the nodes xi and weights...

. A polynomial P ( x ) {\displaystyle P(x)} with coefficients in k {\displaystyle k} is called an additive polynomial, or a Frobenius polynomial, if P...

That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients are integers...

In mathematics, the Conway polynomial Cp,n for the finite field Fpn is a particular irreducible polynomial of degree n over Fp that can be used to define...

solutions, similarly as polynomial algebras are used for the study of algebraic varieties, which are solution sets of systems of polynomial equations. Weyl algebras...

Auxiliary polynomial is a term in mathematics which may refer to: The auxiliary function argument in transcendence theory The characteristic polynomial of a...

In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation ( 1 − x 2 ) d 2 d x 2 P ℓ m ( x ) − 2...

NP-complete. That is, it is in NP, and any problem in NP can be reduced in polynomial time by a deterministic Turing machine to the Boolean satisfiability problem...

Narayana polynomials are a class of polynomials whose coefficients are the Narayana numbers. The Narayana numbers and Narayana polynomials are named after...

mathematics, the Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty...

tensor-based linear solvers. If the state equation is polynomial, then the problem is known as the polynomial–quadratic regulator (PQR). Again, the Al'Brekht...

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 )...

A fully polynomial-time approximation scheme (FPTAS) is an algorithm for finding approximate solutions to function problems, especially optimization problems...

the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that...

In mathematics, the Romanovski polynomials are one of three finite subsets of real orthogonal polynomials discovered by Vsevolod Romanovsky (Romanovski...

the case of polynomials over an arbitrary commutative ring, where p is replaced by an ideal, and "coprime polynomials" means "polynomials that generate...

a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or any other type of expression. It may be a number without...

splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial splits, i.e., decomposes...

power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients...

arrange the coefficients of the polynomial into a square matrix, called the Hurwitz matrix, and showed that the polynomial is stable if and only if the sequence...

is a straight line, that is, a polynomial function of degree zero (a constant polynomial) or one (a linear polynomial). For distinguishing such a linear...

the Neumann polynomials, introduced by Carl Neumann for the special case α = 0 {\displaystyle \alpha =0} , are a sequence of polynomials in 1 / t {\displaystyle...

In mathematics, a Diophantine equation is a polynomial equation with integer coefficients, for which only integer solutions are of interest. A linear Diophantine...

mathematics, an integer-valued polynomial (also known as a numerical polynomial) P ( t ) {\displaystyle P(t)} is a polynomial whose value P ( n ) {\displaystyle...

In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication...

In mathematics, a polynomial p {\displaystyle p} whose Laplacian is zero is termed a harmonic polynomial. The harmonic polynomials form a subspace of the...

Bernstein's theorem is an inequality relating the maximum modulus of a complex polynomial function on the unit disk with the maximum modulus of its derivative on...

theorem connects polynomial factors with polynomial roots. Specifically, if f ( x ) {\displaystyle f(x)} is a (univariate) polynomial, then x − a {\displaystyle...

} where a is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial. A quartic equation, or equation of the fourth degree...

homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial can be expressed as a polynomial expression in complete...

mathematics, a polynomial transformation consists of computing the polynomial whose roots are a given function of the roots of a polynomial. Polynomial transformations...

NP-hard) if there is an algorithm for the problem whose running time is polynomial in the dimension of the problem and the magnitudes of the data involved...

{\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 ∑...

Logarithmic-space Polynomial-time), is the complexity class of computational complexity theory problems solvable in logarithmic space and polynomial time with...

Lill's method is a visual method of finding the real roots of a univariate polynomial of any degree. It was developed by Austrian engineer Eduard Lill in 1867...

{\displaystyle \alpha \in E} , the minimal polynomial of α {\displaystyle \alpha } over F is a separable polynomial (i.e., it is coprime to its formal derivative...

complex quadratic polynomial is a quadratic polynomial whose coefficients and variable are complex numbers. Quadratic polynomials have the following...

is a method to find small integer zeroes of univariate or bivariate polynomials, or their small zeroes modulo a given integer. The method uses the...

of orthogonal polynomials, Bochner's theorem is a characterization theorem of certain families of orthogonal polynomials as polynomial solutions to Sturm–Liouville...

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...

Zolotarev polynomials are polynomials used in approximation theory. They are sometimes used as an alternative to the Chebyshev polynomials where accuracy...

In coding theory, the weight enumerator polynomial of a binary linear code specifies the number of words of each possible Hamming weight. Let C ⊂ F 2 n...

computationally indistinguishable if for any non-uniform probabilistic polynomial time algorithm A, the following quantity is a negligible function in n:...

for all k is if the polynomial function is constant. The same reasoning shows an even stronger result: no non-constant polynomial function P(n) exists...

In mathematics, Heckman–Opdam polynomials (sometimes called Jacobi polynomials) Pλ(k) are orthogonal polynomials in several variables associated to root...

polynomial of degree m − 1 where m ≥ 2 and the coefficient of the highest-order term be a ≠ 0. Assuming the following holds true for all polynomials of...

with the interpretation of choosing 0 elements from a set and simplifies polynomial and binomial expansions. In other contexts, particularly in mathematical...

mathematics, Macdonald-Koornwinder polynomials (also called Koornwinder polynomials) are a family of orthogonal polynomials in several variables, introduced...

The Jenkins–Traub algorithm for polynomial zeros is a fast globally convergent iterative polynomial root-finding method published in 1970 by Michael A...

In mathematics, a linearised polynomial (or q-polynomial) is a polynomial for which the exponents of all the constituent monomials are powers of q and...

(integer division). In algebra of polynomials, the remainder is the polynomial "left over" after dividing one polynomial by another. The modulo operation...

In combinatorial mathematics, the q-difference polynomials or q-harmonic polynomials are a polynomial sequence defined in terms of the q-derivative. They...

case of Chinese remainder theorem for polynomials is Lagrange interpolation. For this, consider k monic polynomials of degree one: P i ( X ) = X − x i ...

In type theory, a polynomial functor (or container functor) is a kind of endofunctor of a category of types that is intimately related to the concept of...

number field sieve is super-polynomial but sub-exponential in the size of the input. Suppose f is a k-degree polynomial over Q {\textstyle \mathbb {Q}...

In mathematics, a twisted polynomial is a polynomial over a field of characteristic p {\displaystyle p} in the variable τ {\displaystyle \tau } representing...

solved for polynomial solutions. Sergei A. Abramov in 1989 and Marko Petkovšek in 1992 described an algorithm which finds all polynomial solutions of...

The polynomial hyperelastic material model is a phenomenological model of rubber elasticity. In this model, the strain energy density function is of the...

bounds the maximum of the derivatives of a polynomial on an interval in terms of the maximum of the polynomial. For k = 1 it was proved by Andrey Markov...

complexity theory, arithmetic circuits are the standard model for computing polynomials. Informally, an arithmetic circuit takes as inputs either variables or...

all decision problems that can be solved by a Turing machine using a polynomial amount of space. If we denote by SPACE(f(n)), the set of all problems...

In mathematics, the Stirling polynomials are a family of polynomials that generalize important sequences of numbers appearing in combinatorics and analysis...

that certain polynomials constructed from the decision variables should be sums of squares. When the maximum degree of the polynomials involved is fixed...

collection of polynomials, and it will have no need to communicate with the central processor until it has finished sieving with its polynomials. The self-initializing...

In mathematics, the Faber polynomials Pm of a Laurent series f ( z ) = z − 1 + a 0 + a 1 z + ⋯ {\displaystyle \displaystyle f(z)=z^{-1}+a_{0}+a_{1}z+\cdots...

above example). Polynomials of degree one are called linear polynomials. Linear algebra studies systems of linear polynomials. A polynomial is said to be...

number of accepting paths of a nondeterministic Turing machine running in polynomial time. Unlike most well-known complexity classes, it is not a class of...

function, is a quadratic polynomial, a polynomial of degree two. In elementary mathematics a polynomial and its associated polynomial function are rarely distinguished...

complexity class NP-easy is the set of function problems that are solvable in polynomial time by a deterministic Turing machine with an oracle for some decision...

has a root of absolute value 1, the minimal polynomial for a Salem number must be a reciprocal polynomial. This implies that 1 / α {\displaystyle 1/\alpha...

for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking,...

A polynomial weir is a weir that has a geometry defined by a polynomial equation of any order n. In practice, most weirs are low-order polynomial weirs...

second levels of the polynomial hierarchy. A language L is in S 2 P {\displaystyle {\mathsf {S}}_{2}^{P}} if there exists a polynomial-time predicate P such...

Chebyshev grid) are a set of specific algebraic numbers used as nodes for polynomial interpolation and numerical integration. They are the projection of a...

polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial...

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...

be the lowest degree term of the polynomial. This also applies to multivariate polynomials. For example, the polynomial x 2 + 2 x y + y 2 − 2 x + 2 y −...

"PP is as Hard as the Polynomial-Time Hierarchy" and was given the 1998 Gödel Prize. The theorem states that the entire polynomial hierarchy PH is contained...

the Bateman polynomials are a family Fn of orthogonal polynomials introduced by Harry Bateman (1933). The Bateman–Pasternack polynomials are a generalization...

However, for polynomials specifically, the study of root-finding algorithms belongs to computer algebra, since algebraic properties of polynomials are fundamental...

a class of cyclic error-correcting codes that are constructed using polynomials over a finite field (also called a Galois field). BCH codes were invented...

In mathematics, Geronimus polynomials may refer to one of the several different families of orthogonal polynomials studied by Yakov Lazarevich Geronimus...

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...

Kostka polynomials Kλμ(q, t) are known by several names including Kostka–Foulkes polynomials, Macdonald–Kostka polynomials or q,t-Kostka polynomials. Here...

In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers { 0 , 1 , 2 , 3 , … } {\textstyle \left\{0,1,2...

mathematics, Carlitz polynomial, named for Leonard Carlitz, may refer to: Al-Salam–Carlitz polynomials Tricomi–Carlitz polynomials This disambiguation...

that polynomial computability with relative error implies polynomial computability with absolute error. The reverse implication, namely that polynomial computability...

In mathematics, a sparse polynomial (also lacunary polynomial or fewnomial) is a polynomial that has far fewer terms than its degree and number of variables...

mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by James Wilson that generalize Jacobi polynomials, Hahn polynomials, and Charlier...

In linear algebra, the Frobenius companion matrix of the monic polynomial p ( x ) = c 0 + c 1 x + ⋯ + c n − 1 x n − 1 + x n {\displaystyle p(x)=c_{0}+c_{1}x+\cdots...

equation for this potential can be described analytically with Gegenbauer polynomials. As v {\displaystyle v} passes through each of these values, a manifold...

to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called...

oblong, kite, rhombus, and square. Four is the highest degree general polynomial equation for which there is a solution in radicals. Four is the only square...

In mathematics, discrete Chebyshev polynomials, or Gram polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced...

deterministic Turing machine in polynomial time (and for which the function problem also represents a predicate decidable in polynomial time). It is the function...

Birkhoff introduced the chromatic polynomial to study the coloring problem, which was generalised to the Tutte polynomial by W. T. Tutte, both of which are...

univariate polynomial p is a sequence of polynomials associated with p and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem...

In mathematics, Charlier polynomials (also called Poisson–Charlier polynomials) are a family of orthogonal polynomials introduced by Carl Charlier in...

is compiled in an off-line phase in order to support some queries in polynomial time. Many ways of compiling a propositional model exist. Different compiled...

polynomial-time algorithm? Does LP admit a strongly polynomial-time algorithm to find a strictly complementary solution? Does LP admit a polynomial-time...

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...

pseudo-polynomial time algorithm using dynamic programming. There is a fully polynomial-time approximation scheme, which uses the pseudo-polynomial time...

answer the question is using knot polynomials, which are invariants of the knot. If two diagrams have different polynomials, they represent different knots...

satisfiability problem (SAT) can be solved by Boolean circuits with a polynomial number of logic gates, then Π 2 = Σ 2 {\displaystyle \Pi _{2}=\Sigma _{2}\...

later found that the flow polynomial is yet another; and soon Tutte discovered an entire class of functions called Tutte polynomials (originally referred to...

UP (unambiguous non-deterministic polynomial-time) is the complexity class of decision problems solvable in polynomial time on an unambiguous Turing machine...

pseudo-polynomial transformation is a function which maps instances of one strongly NP-complete problem into another and is computable in pseudo-polynomial time...

zero test or p/q theorem) states a constraint on rational solutions of a polynomial equation a n x n + a n − 1 x n − 1 + ⋯ + a 0 = 0 {\displaystyle...

recursive method to calculate the coefficients of the characteristic polynomial p A ( λ ) = det ( λ I n − A ) {\displaystyle p_{A}(\lambda )=\det(\lambda...

This solution does not count as polynomial time in complexity theory because B − A {\displaystyle B-A} is not polynomial in the size of the problem, which...

uniquely identifies the function: Algebraic normal form or Zhegalkin polynomial, as a XOR of ANDs of the arguments (no complements allowed) Full (canonical)...

third-degree polynomial with complex coefficients and the zeroes of its derivative. See also geometrical properties of polynomial roots. A cubic polynomial has...

the splitting field of the polynomial f(x) = xq − x. Such a splitting field is an extension of Fp in which the polynomial f has q zeros. This means f...

In mathematics, an all one polynomial (AOP) is a polynomial in which all coefficients are one. Over the finite field of order two, conditions for the AOP...

In computer science, control-flow analysis (CFA) is a static-code-analysis technique for determining the control flow of a program. The control flow is...

such a function is, up to multiplication by a function not zero at P, a polynomial in one fixed variable z, which is monic, and whose coefficients of lower...

complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. More formally, a ring...

mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with...

is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four: P ( x ) = x 4 + a 3 x 3 + a 2 x 2 + a...

relation between the roots of a polynomial P and the roots of its derivative P'. The set of roots of a real or complex polynomial is a set of points in the...

algorithm for computing the polynomial greatest common divisor and the coefficients of Bézout's identity of two univariate polynomials. The extended Euclidean...

analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the polynomial ring may be regarded...

NFL quarterbacks who have posted a passer rating of zero 0 polynomial, a type of polynomial 0 page, or null page, the series of memory addresses at the...

using Taylor polynomials. In cryptography and hash tables, polynomials are used to compute k-independent hashing. In the former case, polynomials are evaluated...

kinds of product; for example, a polynomial p is a multiple of another polynomial q if there exists third polynomial r such that p = qr. 14, 49, −21 and...

as polynomials. P-recursive equations are linear recurrence equations (or linear recurrence relations or linear difference equations) with polynomial coefficients...

Jack polynomial, introduced by Henry Jack. The Jack polynomial is a homogeneous, symmetric polynomial which generalizes the Schur and zonal polynomials, and...

impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here,...

Stieltjes polynomials En are polynomials associated to a family of orthogonal polynomials Pn. They are unrelated to the Stieltjes polynomial solutions...

isolation of a polynomial consist of producing disjoint intervals of the real line, which contain each one (and only one) real root of the polynomial, and, together...

making the Vandermonde matrix invertible. The polynomial interpolation problem is to find a polynomial p ( x ) = a 0 + a 1 x + a 2 x 2 + ⋯ + a n x n {\displaystyle...

In computational complexity theory, Polynomial Local Search (PLS) is a complexity class that models the difficulty of finding a locally optimal solution...

class P/poly consists of Boolean functions computable by circuits of polynomial size. Proving that N P ⊈ P / p o l y {\displaystyle {\mathsf {NP}}\not...

Konhauser polynomials, introduced by Joseph Konhauser, are biorthogonal polynomials for the distribution function of the Laguerre polynomials. Konhauser...

in P". The algorithm was the first one which is able to determine in polynomial time, whether a given number is prime or composite without relying on...

problem with only polynomial overhead. If P is different from co-NP, then all of the co-NP-complete problems are not solvable in polynomial time. If there...

Ruffini's rule is a method for computation of the Euclidean division of a polynomial by a binomial of the form x − r. It was described by Paolo Ruffini in...

inspection problem may be solved in polynomial time, and this duality allows the maximum cut problem to also be solved in polynomial time for planar graphs. The...

set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables...

such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in...

interpolation (sometimes lerp) is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known...

In mathematics, the division polynomials provide a way to calculate multiples of points on elliptic curves and to study the fields generated by torsion...

corresponding to a polynomial is a particular matrix whose nonzero entries are coefficients of the polynomial. Namely, given a real polynomial p ( z ) = a 0...

restricted to homogeneous polynomials of even degree, since a polynomial of odd degree changes sign, and the homogenization of a polynomial takes only nonnegative...

input as a polynomial, but over the Galois field GF(2). Instead of seeing the input as a polynomial of bytes, it is seen as a polynomial of bits, and...

asymptotics for the Hermite polynomial and Laguerre polynomial. Nowadays asymptotic expansions of this kind for orthogonal polynomials are referred to as Plancherel–Rotach...

algebra, completing the square is a technique for converting a quadratic polynomial of the form ⁠ a x 2 + b x + c {\displaystyle \textstyle ax^{2}+bx+c} ⁠...

reformulates the vector of values to be committed as a polynomial. First, we calculate a polynomial such that p ( i ) = x i {\displaystyle p(i)=x_{i}} for...

differential geometry. They can also be defined as implicit equations, often polynomial equations (which spawned algebraic geometry). Analytic geometry also makes...

is known as the indicial polynomial, which is quadratic in r. The general definition of the indicial polynomial is the coefficient of the lowest...

In mathematics, Gottlieb polynomials are a family of discrete orthogonal polynomials given by ℓ n ( x , λ ) = e − n λ ∑ k ( 1 − e λ ) k ( n k ) ( x k...

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 ) =...

Swinnerton-Dyer polynomials are a family of polynomials, introduced by Peter Swinnerton-Dyer, that serve as examples where polynomial factorization algorithms...

the zero set of a polynomial in two indeterminates. More generally, an algebraic curve is the zero set of a finite set of polynomials, which satisfies...

Rabin fingerprinting scheme (aka Polynomial fingerprinting) is a method for implementing fingerprints using polynomials over a finite field. It was proposed...

that the number of spanning trees can be computed from the graph data in polynomial time. Kirchhoff's theorem is a generalization of Cayley's formula which...

of differentiation) because of the symmetry of second derivatives. The polynomial p obtained by replacing partials ∂ ∂ x i {\displaystyle {\frac {\partial...

problem in computer science Can the graph isomorphism problem be solved in polynomial time? More unsolved problems in computer science The graph isomorphism...

minimal polynomial P of a square matrix A is the unique monic polynomial of least degree, m, such that P(A) = 0. Alternatively, the set of polynomials that...

. 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...

Appell polynomials Chebyshev polynomials Difference polynomials Generalized Appell polynomials q-difference polynomials Other sequences generated by more...

In knot theory, the Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman. It is initially defined on a link diagram as F ( K ) ( a...

The Abel polynomials are a sequence of polynomials named after Niels Henrik Abel, defined by the following equation: p n ( x ) = x ( x − a n ) n − 1 {\displaystyle...

homogeneous of weight j {\displaystyle j} : so can be expressed as a polynomial td j ⁡ ( p 1 , … , p j ) {\displaystyle \operatorname {td} _{j}(p_{1}...

organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in Andrews...

solves each SAT problem (where "efficiently" means "deterministically in polynomial time"). Although such an algorithm is generally believed not to exist...

complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known...

concerning the relationship between free modules and projective modules over polynomial rings. In the geometric setting it is a statement about the triviality...

describing the solutions to quintic polynomials. The study of field extensions and their relationship to the polynomials that give rise to them via Galois...

pseudo-polynomial and weakly polynomial is that a pseudo-polynomial bound may be polynomial in U {\displaystyle U} , but for a weakly polynomial bound...

Angelescu polynomials πn(x) are a series of polynomials generalizing the Laguerre polynomials introduced by Aurel Angelescu. The polynomials can be given...

This theorem states that every single-variable polynomial of degree n has n roots. Further, a polynomial with complex coefficients has at least one complex...

complexity class of problems solvable by a deterministic Turing machine in polynomial time (class P) and polylogarithmic space (class PolyL) (that is, O((log...

roots of a Chebyshev polynomial and these values are used to construct a polynomial approximation for the function. This polynomial is then integrated exactly...

\,} The Alexander polynomial and Conway polynomial are the same as those for the knot 946, but the Jones polynomials for these two knots are different...

scaling the H ( s ) {\displaystyle H(s)} polynomials to the inverse of that frequency. To scale the polynomials, simply append ω c {\displaystyle \omega...

for polynomial interpolation that was derived by the mathematician Eric Harold Neville in 1934. Given n + 1 points, there is a unique polynomial of degree...

extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry...

The Polynomial: Space of the Music is a 3D shooter music and indie game by Russian developer Dmytry Lavrov, released on May 28, 2009. Allen, David (September...

In mathematics, a system of linear equations or a system of polynomial equations is considered underdetermined if there are fewer equations than unknowns...

series that fits the data. The resulting polynomial may be used to extrapolate the data. High-order polynomial extrapolation must be used with due care...

in computer algebra and number theory. A simple concept is that of a polynomial vector field, in other words a vector field expressed with respect to...

an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize...

In mathematics, the Peters polynomials sn(x) are polynomials studied by George Peters given by the generating function ∑ n = 0 + ∞ s n ( x ) t n n ! =...

class, a non-uniform analogue of the class NP of problems solvable in polynomial time by a non-deterministic Turing machine. It is the non-deterministic...

over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard...

L^{-1}X_{t}=X_{t+1}} and L k X t = X t − k . {\displaystyle L^{k}X_{t}=X_{t-k}.} Polynomials of the lag operator can be used, and this is a common notation for ARMA...

set of decision problems solvable by a deterministic Turing machine in polynomial time. There are, however, many complexity classes defined in terms of...

Bézout's theorem is a statement concerning the number of common zeros of n polynomials in n indeterminates. In its original form the theorem states that in...

represented as a product involving its zeroes and an exponential of a polynomial. It is named for Jacques Hadamard. The theorem may be viewed as an extension...

orthogonal polynomials (MOPs) are orthogonal polynomials in one variable that are orthogonal with respect to a finite family of measures. The polynomials are...

division of polynomials, with less writing and fewer calculations than long division. It is mostly taught for division by linear monic polynomials (known as...

there are many algorithms for finding the optimal assignment in time polynomial in n. The assignment problem is a special case of the transportation problem...

In mathematics, Sobolev orthogonal polynomials are orthogonal polynomials with respect to a Sobolev inner product, i.e. an inner product with derivatives...

description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely...

secondary polynomials { q n ( x ) } {\displaystyle \{q_{n}(x)\}} associated with a sequence { p n ( x ) } {\displaystyle \{p_{n}(x)\}} of polynomials orthogonal...

Chihara polynomials may refer to one of the families of orthogonal polynomials studied by Theodore Seio Chihara, including Al-Salam–Chihara polynomials Brenke–Chihara...

between Shabat polynomials and Chebyshev polynomials, Shabat polynomials themselves are sometimes called generalized Chebyshev polynomials. Different trees...

of Legendre's differential equation. The Legendre polynomials and the associated Legendre polynomials are also solutions of the differential equation in...

In algebra, a sextic (or hexic) polynomial is a polynomial of degree six. A sextic equation is a polynomial equation of degree six—that is, an equation...

{\displaystyle I(X)} is the ideal of all polynomials vanishing on X {\displaystyle X} . For any set of polynomials S {\displaystyle S} , let T {\displaystyle...

by means of the minimal polynomial of 2 cos ⁡ ( 2 π / n ) . {\displaystyle 2\cos(2\pi /n).} The roots of the minimal polynomial are twice the real part...

frequently a polynomial over the integers may evaluate to a multiple of a fixed prime p. More precisely, it states that for all integer polynomials f ∈ Z [...

presumed difficulty of factoring certain polynomials in a truncated polynomial ring into a quotient of two polynomials having very small coefficients. Breaking...

certain Jones polynomials, and the quantum algorithm for linear systems of equations, have quantum algorithms appearing to give super-polynomial speedups and...

poweroid is a polynomial sequence, i.e., a sequence ( pn(x) : n = 0, 1, 2, 3, ... ) of polynomials in which the index of each polynomial equals its degree...

bound) is one of a family of inequalities concerning a univariate integer polynomial f(x) and one of its factors h(x). A basic version states that the coefficients...

advanced analog machines that could solve real and complex roots of polynomials, which were published in 1901 by the Paris Academy of Sciences. Charles...

equations. A linear function is a polynomial function in which the variable x has degree at most one (a linear polynomial): f ( x ) = a x + b {\displaystyle...

In mathematics, the actuarial polynomials a(β) n(x) are polynomials given by the generating function ∑ n a n ( β ) ( x ) n ! t n = exp ⁡ ( β t + x ( 1...

A non-constant polynomial with coefficients in a field is said to be eventually stable if the number of irreducible factors of the n {\displaystyle n}...

Xuong tree and an associated maximum-genus embedding can be found in polynomial time. A tree is a connected undirected graph with no cycles. It is a spanning...

golden ratio is a root of a polynomial with rational coefficients, it is an algebraic number. Its minimal polynomial, the polynomial of lowest degree with integer...

extending the field F (whence the "rational"), notably without factoring polynomials, this shows that whether two matrices are similar does not change upon...

with the best polynomial approximation of the function (the degree of the polynomials are fixed). The Lebesgue constant for polynomials of degree at most...

special Schubert cycle in the Schubert calculus, or the product of a Schur polynomial by a complete symmetric function. In terms of Schur functions sλ indexed...

interpolation attack expresses the ciphertext as a polynomial of the plaintext. If the polynomial has a relative low number of unknown coefficients, then...

mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence { p n ( x ) } n = 0 , 1 , 2 , … {\displaystyle \{p_{n}(x)\}_{n=0...

In mathematics, the Boole polynomials sn(x) are polynomials given by the generating function ∑ s n ( x ) t n / n ! = ( 1 + t ) x 1 + ( 1 + t ) λ {\displaystyle...

 zn−1, zn = z0 = 1 are all of the nth roots of unity, since an nth-degree polynomial equation over a field (in this case the field of complex numbers) has...

to 0 a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence. The polynomial's linearity...

The Bollobás–Riordan polynomial can mean a 3-variable invariant polynomial of graphs on orientable surfaces, or a more general 4-variable invariant of...

Chebyshev polynomials, Charlier polynomials, Krawtchouk polynomials, Meixner polynomials, dual Hahn polynomials, Hahn polynomials, and Racah polynomials. If...

factorial, falling sequential product, or lower factorial) is defined as the polynomial ( x ) n = x n _ = x ( x − 1 ) ( x − 2 ) ⋯ ( x − n + 1 ) ⏞ n  factors =...

−1.) This polynomial helps to solve some basic questions; see below. Another polynomial associated with A is the Whitney-number polynomial wA(x, y), defined...

{\textstyle k} "ascents"). Leonhard Euler investigated them and associated polynomials in his 1755 book Institutiones calculi differentialis. Other notations...

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...

In mathematics, and especially gauge theory, Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces of anti-self-dual...

itself. A decision problem is in the complexity class P/f(n) if there is a polynomial time Turing machine M with the following property: for any n, there is...

multi-homogeneous polynomials of Bézout's theorem, which counts the number of isolated common zeros of a set of homogeneous polynomials. This generalization...

expand Macdonald polynomials in terms of LLT polynomials. Ian Grojnowski and Mark Haiman proved a positivity conjecture for LLT polynomials that combined...

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...

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...

high-degree polynomial: if the polynomial evaluates above zero, that point is classified as positive, otherwise as negative. A high-degree polynomial can be...

of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683. Simply, it is a method for transforming a polynomial equation of degree...

symmetric polynomials in n indeterminates, as n goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can...

In mathematics, the Hall–Littlewood polynomials are symmetric functions depending on a parameter t and a partition λ. They are Schur functions when t...

one can see a connection between Fibonacci anyons and the Jones polynomial polynomial using the classical techniques of Edward Witten. In his seminal...

over Q, or of a polynomial, regarded as a vector of coefficients, or of an algebraic number, from the height of its minimal polynomial. The naive height...

In mathematics, Racah polynomials are orthogonal polynomials named after Giulio Racah, as their orthogonality relations are equivalent to his orthogonality...

In theoretical computer science, almost wide probabilistic polynomial-time (AWPP) is a complexity class contained in PP defined via GapP functions. The...

conjecture is a famous unsolved problem concerning polynomials in several variables. It states that if a polynomial function from an n-dimensional space to itself...

is an algorithm for numerical evaluation of polynomials. Horner's method for evaluation of polynomials is one of the most commonly used algorithms for...

any. The solutions of homogeneous linear differential equations with polynomial coefficients are called holonomic functions. This class of functions is...

then a polynomial (in the variables xij, i < j ): this coincides with the square of the Pfaffian of the matrix A and is non-zero (as a polynomial) if and...

Birkhoff interpolation is an extension of polynomial interpolation. It refers to the problem of finding a polynomial P ( x ) {\displaystyle P(x)} of degree...

relationship between polynomials and polynomial functions, the term "constant" is often used to denote the coefficients of a polynomial, which are constant...

and a is nonzero. In other words, a quintic function is defined by a polynomial of degree five. Because they have an odd degree, normal quintic functions...

of abstract algebra, a radical polynomial is a multivariate polynomial over a field that can be expressed as a polynomial in the sum of squares of the variables...

provide a general algorithm that, for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns), can...

In algebra, the ring of polynomial differential forms on the standard n-simplex is the differential graded algebra: Ω poly ∗ ( [ n ] ) = Q [ t 0 , . ....

Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890, that were introduced...

cochain complex. It may be regarded as a categorification of the Jones polynomial. It was developed in the late 1990s by Mikhail Khovanov. To any link diagram...

interpolating polynomial Sir George Biddell Airy: Airy function Waleed Al-Salam (1926–1996): Al-Salam polynomial - Al Salam–Carlitz polynomial - Al Salam–Chihara...

converting between sample values and the coefficients of a trigonometric polynomial that interpolates those values. It is therefore a basic tool for numerical...

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...

polynomials, Buchberger's algorithm is a method for transforming a given set of polynomials into a Gröbner basis, which is another set of polynomials...

can be solved in polynomial time, but does not provide a concrete polynomial-time algorithm for solving it. Such proofs of polynomiality are non-constructive:...

accept. The algorithm can check in polynomial time if the vertices in G appear once in c. Additionally, it takes polynomial time to check the start and end...

mathematics, trigonometric interpolation is interpolation with trigonometric polynomials. Interpolation is the process of finding a function which goes through...

is simply the larger learning with errors (LWE) problem specialized to polynomial rings over finite fields. Because of the presumed difficulty of solving...

mutation to the Conway knot, with which it shares a Jones polynomial. It has the same Alexander polynomial as the unknot. Weisstein, Eric W. "Conway's Knot"....

transcendental number, which means that it is not the solution of any non-constant polynomial equation with rational coefficients, such as x 5 120 − x 3 6 + x = 0 {\textstyle...

In mathematics, a Dirichlet series is any series of the form ∑ n = 1 ∞ a n n s , {\displaystyle \sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}},} where s is...

is the set of problems solvable by an interactive proof system with a polynomial-time verifier and one computationally unbounded prover. Informally, IP...

In mathematics the Mott polynomials sn(x) are polynomials given by the exponential generating function: e x ( 1 − t 2 − 1 ) / t = ∑ n s n ( x ) t n /...

over Q {\displaystyle \mathbb {Q} } by using the fact that a symmetric polynomial whose arguments are all conjugates of one another gives a rational number...

In 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...

power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to...

H_{f}:\mathbf {C} [z]\to \mathbf {z} ^{-1}\mathbf {C} [[z^{-1}]].} This takes a polynomial g ∈ C [ z ] {\displaystyle g\in \mathbf {C} [z]} and sends it to the product...

theorem states that bounded-error probabilistic polynomial (BPP) time is contained in the polynomial time hierarchy, and more specifically Σ2 ∩ Π2. In...

knot. Its Alexander polynomial is Δ ( t ) = 3 t − 5 + 3 t − 1 , {\displaystyle \Delta (t)=3t-5+3t^{-1},\,} its Conway polynomial is ∇ ( z ) = 3 z 2 +...

conditions for the existence of solutions to systems of multivariate polynomial equations over an algebraically closed field (such as the complex numbers...

value of function's best approximation by algebraic or trigonometric polynomials in terms of the modulus of continuity or modulus of smoothness of the...

decomposition is an algorithm that allows testing whether a system of polynomial equations and inequalities has solutions, and, if solutions exist, describing...

or because of its Conway polynomial, which is ∇ ( z ) = z 2 + 1. {\displaystyle \nabla (z)=z^{2}+1.} The Jones polynomial is V ( q ) = q − 1 + q − 3...

b x 2 + c x + d , {\displaystyle f(x)=ax^{3}+bx^{2}+cx+d,} that is, a polynomial function of degree three. In many texts, the coefficients a, b, c, and...

then it has at least one real root (this is true for all odd-degree polynomial functions). All of the roots of the cubic equation can be found by the...

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)...

function ex not with a polynomial but with a rational function, that is a quotient of two polynomials. In particular he chose polynomials A(x) and B(x) such...

calculate knot invariants and three-manifold invariants such as the Jones polynomial. Particularly, Chern–Simons theory is specified by a choice of simple...

polynomial time. There are two kinds of time complexity results: Positive results show that a certain class of functions can be learned in polynomial...

potentially diagonalisable and the other is nilpotent. The two parts are polynomials in the operator, which makes them behave nicely in algebraic manipulations...

edges. In contrast to the shortest path problem, which can be solved in polynomial time in graphs without negative-weight cycles, the longest path problem...

operator. The denominator is a Butterworth polynomial in s {\displaystyle s} . The Butterworth polynomials may be written in complex form as above, but...

combinatorial optimization algorithm that solves the assignment problem in polynomial time and which anticipated later primal–dual methods. It was developed...

Bunyakovsky conjecture (or Bouniakowsky conjecture) gives a criterion for a polynomial f ( x ) {\displaystyle f(x)} in one variable with integer coefficients...

the time. Thus, Arthur acts as a probabilistic polynomial-time verifier, assuming it is allotted polynomial time to make its decisions and queries. The simplest...

function, denoted Hf, if it is also regular at z = 1, and is called a Heun polynomial, denoted Hp, if it is regular at all three finite singular points z = 0...

{1-x^{2}}{1+x^{2}}}}} An algebraic equation is an equation involving polynomials, for which algebraic expressions may be solutions. If the set of constants...

to eliminating some variables between polynomials of several variables, in order to solve systems of polynomial equations. Classical elimination theory...

In mathematics, the Pidduck polynomials sn(x) are polynomials introduced by Pidduck given by the generating function ∑ n s n ( x ) n ! t n = ( 1 + t 1...

In computational complexity theory, randomized polynomial time (RP) is the complexity class of decision problems for which a probabilistic Turing machine...

technique for solving differential equations by converting them into polynomial equations. The central idea is to view integration and differentiation...

(e.g. reservoir flow-rates) There is a large amount of literature on polynomial-time algorithms for certain special classes of discrete optimization....

formula. The dimension of a Demazure module is a polynomial in the highest weight, called a Demazure polynomial. Suppose that g is a complex semisimple Lie...

extension are called Ore polynomials. Ore extensions appear in several natural contexts, including skew and differential polynomial rings, group algebras...

subset U ⊂ Rn is an analytic function f: U → R satisfying a nontrivial polynomial equation P(x,f(x)) = 0 for all x in U (A semialgebraic subset of Rn is...

types of algebraic geometry), the Chern classes can be expressed as polynomials in the coefficients of the curvature form. There are various ways of...

\operatorname {S} _{n}(x)} only for integers n ≥ 1. The order of the polynomial in the general smoothstep is 2n + 1. With n = 1, the slopes or first derivatives...

system exhibiting a lemniscate shape Polynomial lemniscate, a level set of the absolute value of a complex polynomial "lemniscate". Dictionary.com Unabridged...

"A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time". Foundations of Computational Mathematics. to...