# Category: Polynomials

Perron's irreducibility criterion
Perron's irreducibility criterion is a sufficient condition for a polynomial to be irreducible in —that is, for it to be unfactorable into the product of lower-degree polynomials with integer coeffici
Jones polynomial
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 of an oriented knot or link which assigns to e
Neumann polynomial
In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case , are a sequence of polynomials in used to expand functions in term of Bessel functions. The first few polynomi
Wilkinson's polynomial
In numerical analysis, Wilkinson's polynomial is a specific polynomial which was used by James H. Wilkinson in 1963 to illustrate a difficulty when finding the root of a polynomial: the location of th
Series expansion
In mathematics, a series expansion is an expansion of a function into a series, or infinite sum. It is a method for calculating a function that cannot be expressed by just elementary operators (additi
Generalized Appell polynomials
In mathematics, a polynomial sequence has a generalized Appell representation if the generating function for the polynomials takes on a certain form: where the generating function or kernel is compose
Faber polynomials
In mathematics, the Faber polynomials Pm of a Laurent series are the polynomials such that vanishes at z=0. They were introduced by Faber and studied by Grunsky and Schur.
Polynomial Wigner–Ville distribution
In signal processing, the polynomial Wigner–Ville distribution is a quasiprobability distribution that generalizes the Wigner distribution function. It was proposed by Boualem Boashash and Peter O'She
Indeterminate (variable)
In mathematics, particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else except itself. It may be used as a placeholder in objec
Cyclic redundancy check
A cyclic redundancy check (CRC) is an error-detecting code commonly used in digital networks and storage devices to detect accidental changes to digital data. Blocks of data entering these systems get
Kauffman polynomial
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 , where is the writhe of the link diagram and is a polynomia
Gould polynomials
In mathematics the Gould polynomials Gn(x; a,b) are polynomials introduced by H. W. Gould and named by Roman in 1984.They are given by where so
Alexander polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomi
Lommel polynomial
A Lommel polynomial Rm,ν(z), introduced by Eugen von Lommel, is a polynomial in 1/z giving the recurrence relation where Jν(z) is a Bessel function of the first kind. They are given explicitly by
Cohn's irreducibility criterion
Arthur Cohn's irreducibility criterion is a sufficient condition for a polynomial to be irreducible in —that is, for it to be unfactorable into the product of lower-degree polynomials with integer coe
Minimal polynomial (field theory)
In field theory, a branch of mathematics, the minimal polynomial of an element α of a field is, roughly speaking, the polynomial of lowest degree having coefficients in the field, such that α is a roo
Mott polynomials
In mathematics the Mott polynomials sn(x) are polynomials introduced by N. F. Mott who applied them to a problem in the theory of electrons. They are given by the exponential generating function Becau
Minimal polynomial (linear algebra)
In linear algebra, the minimal polynomial μA of an n × n matrix A over a field F is the monic polynomial P over F of least degree such that P(A) = 0. Any other polynomial Q with Q(A) = 0 is a (polynom
Peters polynomials
In mathematics, the Peters polynomials sn(x) are polynomials studied by Peters given by the generating function , . They are a generalization of the Boole polynomials.
Polynomial decomposition
In mathematics, a polynomial decomposition expresses a polynomial f as the functional composition of polynomials g and h, where g and h have degree greater than 1; it is an algebraic functional decomp
LLT polynomial
In mathematics, an LLT polynomial is one of a family of symmetric functions introduced by Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon (1997) as q-analogues of products of Schur functions. J.
Sextic equation
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 whose left hand side is a sextic polynomial and
Chebyshev polynomials
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined in several equivalent ways, one of which starts with trigonome
Polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indetermin
Matrix polynomial
In mathematics, a matrix polynomial is a polynomial with square matrices as variables. Given an ordinary, scalar-valued polynomial this polynomial evaluated at a matrix A is where I is the identity ma
Quasi-polynomial
In mathematics, a quasi-polynomial (pseudo-polynomial) is a generalization of polynomials. While the coefficients of a polynomial come from a ring, the coefficients of quasi-polynomials are instead pe
Discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynom
Hermite polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probabili
Lill's method
In mathematics, 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. A later paper by Lill de
Appell sequence
In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence satisfying the identity and in which is a non-zero constant. Among the most notable Appell sequences besid
Humbert polynomials
In mathematics, the Humbert polynomials πλn,m(x) are a generalization of Pincherle polynomials introduced by Humbert given by the generating function , p.58).
Ring of symmetric functions
In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity. This ring
Touchard polynomials
The Touchard polynomials, studied by Jacques Touchard, also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence of binomial type defined by where is a Stirling numbe
Sylvester matrix
In mathematics, a Sylvester matrix is a matrix associated to two univariate polynomials with coefficients in a field or a commutative ring. The entries of the Sylvester matrix of two polynomials are c
Fekete polynomial
In mathematics, a Fekete polynomial is a polynomial where is the Legendre symbol modulo some integer p > 1. These polynomials were known in nineteenth-century studies of Dirichlet L-functions, and ind
Remez algorithm
The Remez algorithm or Remez exchange algorithm, published by Evgeny Yakovlevich Remez in 1934, is an iterative algorithm used to find simple approximations to functions, specifically, approximations
Laurent polynomial
In mathematics, a Laurent polynomial (namedafter Pierre Alphonse Laurent) in one variable over a field is a linear combination of positive and negative powers of the variable with coefficients in . La
Linearised polynomial
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 the coefficients come from some extension field o
Mahler polynomial
In mathematics, the Mahler polynomials gn(x) are polynomials introduced by Mahler in his work on the zeros of the incomplete gamma function. Mahler polynomials are given by the generating function Mah
Rosenbrock function
In mathematical optimization, the Rosenbrock function is a non-convex function, introduced by Howard H. Rosenbrock in 1960, which is used as a performance test problem for optimization algorithms. It
Linear relation
In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution. More precisely, if are elements of
Bernstein–Sato polynomial
In mathematics, the Bernstein–Sato polynomial is a polynomial related to differential operators, introduced independently by Joseph Bernstein and Mikio Sato and Takuro Shintani , . It is also known as
Algebraic equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form where P is a polynomial with coefficients in some field, often the field of the rational numbers. For many autho
Bernoulli polynomials of the second kind
The Bernoulli polynomials of the second kind ψn(x), also known as the Fontana-Bessel polynomials, are the polynomials defined by the following generating function: The first five polynomials are: Some
Eisenstein's criterion
In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into
Irreducible polynomial
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nat
Polynomial expansion
In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtaine
Trinomial
In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials.
Kharitonov region
A Kharitonov region is a concept in mathematics. It arises in the study of the stability of polynomials. Let be a simply-connected set in the complex plane and let be the polynomial family. is said to
Heine–Stieltjes polynomials
In mathematics, the Heine–Stieltjes polynomials or Stieltjes polynomials, introduced by T. J. Stieltjes, are polynomial solutions of a second-order Fuchsian equation, a differential equation all of wh
Polynomial greatest common divisor
In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This
Polynomial matrix spectral factorization
Polynomial matrices are widely studied in the fields of systems theory and control theory and have seen other uses relating to stable polynomials. In stability theory, Spectral Factorization has been
Caloric polynomial
In differential equations, the mth-degree caloric polynomial (or heat polynomial) is a "parabolically m-homogeneous" polynomial Pm(x, t) that satisfies the heat equation "Parabolically m-homogeneous"
Kazhdan–Lusztig polynomial
In the mathematical field of representation theory, a Kazhdan–Lusztig polynomial is a member of a family of integral polynomials introduced by David Kazhdan and George Lusztig. They are indexed by pai
Vandermonde polynomial
In algebra, the Vandermonde polynomial of an ordered set of n variables , named after Alexandre-Théophile Vandermonde, is the polynomial: (Some sources use the opposite order , which changes the sign
Neville's algorithm
In mathematics, Neville's algorithm is an algorithm used for polynomial interpolation that was derived by the mathematician Eric Harold Neville in 1934. Given n + 1 points, there is a unique polynomia
Order polynomial
The order polynomial is a polynomial studied in mathematics, in particular in algebraic graph theory and algebraic combinatorics. The order polynomial counts the number of order-preserving maps from a
Lebesgue constant
In mathematics, the Lebesgue constants (depending on a set of nodes and of its size) give an idea of how good the interpolant of a function (at the given nodes) is in comparison with the best polynomi
Ehrhart polynomial
In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The the
Difference polynomials
In mathematics, in the area of complex analysis, the general difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selb
Integer-valued polynomial
In mathematics, an integer-valued polynomial (also known as a numerical polynomial) is a polynomial whose value is an integer for every integer n. Every polynomial with integer coefficients is integer
Tian yuan shu
Tian yuan shu (simplified Chinese: 天元术; traditional Chinese: 天元術; pinyin: tiān yuán shù) is a Chinese system of algebra for polynomial equations. Some of the earliest existing writings were created in
Primitive polynomial (field theory)
In finite field theory, a branch of 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 with
Hudde's rules
In mathematics, Hudde's rules are two properties of polynomial roots described by Johann Hudde. 1. If r is a double root of the polynomial equation and if are numbers in arithmetic progression, then r
Characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of
Quasisymmetric function
In algebra and in particular in algebraic combinatorics, a quasisymmetric function is any element in the ring of quasisymmetric functions which is in turn a subring of the formal power series ring wit
Secondary polynomials
In mathematics, the secondary polynomials associated with a sequence of polynomials orthogonal with respect to a density are defined by To see that the functions are indeed polynomials, consider the s
Zolotarev polynomials
In mathematics, Zolotarev polynomials are polynomials used in approximation theory. They are sometimes used as an alternative to the Chebyshev polynomials where accuracy of approximation near the orig
Sheffer sequence
In mathematics, a Sheffer sequence or 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, satisfyi
In mathematics, in the realm of abstract algebra, a radial polynomial is a multivariate polynomial over a field that can be expressed as a polynomial in the sum of squares of the variables. That is, i
Horner's method
In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has be
Rainville polynomials
In mathematics, the Rainville polynomials pn(z) are polynomials introduced by given by the generating function , p.46).
Root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches of m
Symmetric polynomial
In mathematics, a 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, P is a symmetric
Polynomial interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points of the dataset. Given a set of n + 1
Actuarial polynomials
In mathematics, the actuarial polynomials a(β)n(x) are polynomials studied by given by the generating function , .
Alternating polynomial
In algebra, an alternating polynomial is a polynomial such that if one switches any two of the variables, the polynomial changes sign: Equivalently, if one permutes the variables, the polynomial chang
Matching polynomial
In the mathematical fields of graph theory and combinatorics, a matching polynomial (sometimes called an acyclic polynomial) is a generating function of the numbers of matchings of various sizes in a
Permutation polynomial
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 is a bijection. In case the ring is a finite field, the
Bernstein polynomial
In the mathematical field of numerical analysis, a Bernstein polynomial is a polynomial that is a linear combination of Bernstein basis polynomials. The idea is named after Sergei Natanovich Bernstein
Height function
A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typica
Quartic equation
In mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is where a ≠ 0. The quartic is the highest order polynomial
Lagrange polynomial
In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a data set of coordinate pairs with the are called n
Reciprocal polynomial
In algebra, given a polynomial with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial, denoted by p∗ or pR, is the polynomial That is, the coefficients of p∗ are
Synthetic division
In algebra, synthetic division is a method for manually performing Euclidean division of polynomials, with less writing and fewer calculations than long division. It is mostly taught for division by l
Polynomial identity testing
In mathematics, polynomial identity testing (PIT) is the problem of efficiently determining whether two multivariate polynomials are identical. More formally, a PIT algorithm is given an arithmetic ci
Degree of a polynomial
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 degree of a term is the sum of the exponents o
Algebraic set
No description available.
Hurwitz polynomial
In mathematics, a Hurwitz polynomial, named after Adolf Hurwitz, is a polynomial whose roots (zeros) are located in the left half-plane of the complex plane or on the imaginary axis, that is, the real
Tutte polynomial
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 an important role in graph theory. It is define
System of polynomial equations
A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations f1 = 0, ..., fh = 0 where the fi are polynomials in several variables, say x1, ..., xn, over
Polynomial sequence
In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Poly
Narumi polynomials
In mathematics, the Narumi polynomials sn(x) are polynomials introduced by given by the generating function ,
In algebra, the Bring radical or ultraradical of a real number a is the unique real root of the polynomial The Bring radical of a complex number a is either any of the five roots of the above polynomi
Septic equation
In algebra, a septic equation is an equation of the form where a ≠ 0. A septic function is a function of the form where a ≠ 0. In other words, it is a polynomial of degree seven. If a = 0, then f is a
Monic polynomial
In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore,
Matrix factorization of a polynomial
In mathematics, a matrix factorization of a polynomial is a technique for factoring irreducible polynomials with matrices. David Eisenbud proved that every multivariate real-valued polynomial p withou
Casus irreducibilis
In algebra, casus irreducibilis (Latin for "the irreducible case") is one of the cases that may arise in solving polynomials of degree 3 or higher with integer coefficients algebraically (as opposed t
Continuant (mathematics)
In algebra, the continuant is a multivariate polynomial representing the determinant of a tridiagonal matrix and having applications in generalized continued fractions.
Mahler measure
In mathematics, the Mahler measure of a polynomial with complex coefficients is defined as where factorizes over the complex numbers as The Mahler measure can be viewed as a kind of height function. U
Routh–Hurwitz stability criterion
In control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stability of a linear time-invariant (LTI) dynamical system
Factorization of polynomials over finite fields
In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically possible and is unique for p
Legendre polynomials
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of ma
Tschirnhaus transformation
In mathematics, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683. Simply, it is a m
Exponential polynomial
In mathematics, exponential polynomials are functions on fields, rings, or abelian groups that take the form of polynomials in a variable and an exponential function.
Cyclotomic polynomial
In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor of and is not a divisor of for any k < n. It
Wall polynomial
In mathematics, a Wall polynomial is a polynomial studied by in his work on conjugacy classes in classical groups, and named by .
List of polynomial topics
Carlitz–Wan conjecture
In mathematics, the Carlitz–Wan conjecture classifies the possible degrees of exceptional polynomials over a finite field Fq of q elements. A polynomial f(x) in Fq[x] of degree d is called exceptional
Symmetric algebra
In mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is a commutative algebra over K that contains V, and is, in some sense, minimal for this property. H
Triangular decomposition
In computer algebra, a triangular decomposition of a polynomial system S is a set of simpler polynomial systems S1, ..., Se such that a point is a solution of S if and only if it is a solution of one
Real-root isolation
In mathematics, and, more specifically in numerical analysis and computer algebra, real-root isolation of a polynomial consist of producing disjoint intervals of the real line, which contain each one
Monomial basis
In mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists of all monomials. The monomials form a basis
Boole polynomials
In mathematics, the Boole polynomials sn(x) are polynomials given by the generating function , .
Equally spaced polynomial
An equally spaced polynomial (ESP) is a polynomial used in finite fields, specifically GF(2) (binary). An s-ESP of degree sm can be written as: for or
Unimodular polynomial matrix
In mathematics, a unimodular polynomial matrix is a square polynomial matrix whose inverse exists and is itself a polynomial matrix. Equivalently, a polynomial matrix A is unimodular if its determinan
Dickson polynomial
In mathematics, the Dickson polynomials, denoted Dn(x,α), form a polynomial sequence introduced by L. E. Dickson. They were rediscovered by in his study of Brewer sums and have at times, although rare
Knot polynomial
In 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 of a given knot.
Bollobás–Riordan polynomial
The Bollobás–Riordan polynomial can mean a 3-variable invariant polynomial of graphs on orientable surfaces, or a more general 4-variable invariant of ribbon graphs, generalizing the Tutte polynomial.
Pidduck polynomials
In mathematics, the Pidduck polynomials sn(x) are polynomials introduced by Pidduck given by the generating function ,
Regular semi-algebraic system
In computer algebra, a regular semi-algebraic system is a particular kind of triangular system of multivariate polynomials over a real closed field.
Samuelson–Berkowitz algorithm
In mathematics, the Samuelson–Berkowitz algorithm efficiently computes the characteristic polynomial of an matrix whose entries may be elements of any unital commutative ring. Unlike the Faddeev–LeVer
Resolvent cubic
In algebra, a resolvent cubic is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four: In each case: * The coefficients of the resolvent cubic c
Fibonacci polynomials
In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numb
Lindsey–Fox algorithm
The Lindsey–Fox algorithm, named after Pat Lindsey and Jim Fox, is a numerical algorithm for finding the roots or zeros of a high-degree polynomial with real coefficients over the complex field. It is
Stanley symmetric function
In mathematics and especially in algebraic combinatorics, the Stanley symmetric functions are a family of symmetric functions introduced by Richard Stanley in his study of the symmetric group of permu
In mathematics, the additive polynomials are an important topic in classical algebraic number theory.
Berlekamp–Rabin algorithm
In number theory, Berlekamp's root finding algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials over a field . The method was discovered by
Pincherle polynomials
In mathematics, the Pincherle polynomials Pn(x) are polynomials introduced by S. Pincherle given by the generating function Humbert polynomials are a generalization of Pincherle polynomials
Polynomial long division
In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of the familiar arithmetic technique called long
Minimal polynomial of 2cos(2pi/n)
For an integer , the minimal polynomial of is the non-zero monic polynomial of degree for and degree for with integer coefficients, such that . Here denotes the Euler's totient function. In particular
Romanovski polynomials
In mathematics, the Romanovski polynomials are one of three finite subsets of real orthogonal polynomials discovered by Vsevolod Romanovsky (Romanovski in French transcription) within the context of p
Hilbert's Nullstellensatz
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and pos
Sister Celine's polynomials
In mathematics, Sister Celine's polynomials are a family of hypergeometric polynomials introduced by Mary Celine Fasenmyer. They include Legendre polynomials, Jacobi polynomials, and Bateman polynomia
Polylogarithmic function
In mathematics, a polylogarithmic function in n is a polynomial in the logarithm of n, The notation logkn is often used as a shorthand for (log n)k, analogous to sin2θ for (sin θ)2. In computer scienc
Multilinear polynomial
In algebra, a multilinear polynomial is a multivariate polynomial that is linear (meaning affine) in each of its variables separately, but not necessarily simultaneously. It is a polynomial in which n
Enumerator polynomial
In coding theory, the weight enumerator polynomial of a binary linear code specifies the number of words of each possible Hamming weight. Let be a binary linear code length . The weight distribution i
Umbral calculus
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain "shadowy" techniques used to "prove" them.
Separable polynomial
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 roots is equal to the degree of the polynomi
Hilbert's thirteenth problem
Hilbert's thirteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It entails proving whether a solution exists for all 7th-degree equations
In mathematics (linear algebra), the Faddeev–LeVerrier algorithm is a recursive method to calculate the coefficients of the characteristic polynomial of a square matrix, A, named after Dmitry Konstant
Rook polynomial
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 a checkerboard; that is, no two rooks may be in
Legendre moment
In mathematics, Legendre moments are a type of image moment and are achieved by using the Legendre polynomial. Legendre moments are used in areas of image processing including: pattern and object reco
Binomial type
In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers in which the index of each polynomial equals its degree, is said to be of binomial type if it sa
Bézout matrix
In mathematics, a Bézout matrix (or Bézoutian or Bezoutiant) is a special square matrix associated with two polynomials, introduced by James Joseph Sylvester and Arthur Cayley and named after Étienne
Monomial order
In mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all (monic) monomials in a given polynomial ring, satisfying the property of resp
All one polynomial
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 to be irreducible are known, which allow this
Stirling polynomials
In mathematics, the Stirling polynomials are a family of polynomials that generalize important sequences of numbers appearing in combinatorics and analysis, which are closely related to the Stirling n
Thomae's formula
In mathematics, Thomae's formula is a formula introduced by Carl Johannes Thomae relating theta constants to the branch points of a hyperelliptic curve .
Maximum length sequence
A maximum length sequence (MLS) is a type of pseudorandom binary sequence. They are bit sequences generated using maximal linear-feedback shift registers and are so called because they are periodic an
Aitken interpolation
Aitken interpolation is an algorithm used for polynomial interpolation that was derived by the mathematician Alexander Aitken. It is similar to Neville's algorithm. See also Aitken's delta-squared pro
FGLM algorithm
FGLM is one of the main algorithms in computer algebra, named after its designers, Faugère, Gianni, Lazard and Mora. They introduced their algorithm in 1993. The input of the algorithm is a Gröbner ba
Morley–Wang–Xu element
In applied mathematics, the Morlely–Wang–Xu (MWX) element is a canonical construction of a family of piecewise polynomials with the minimal degree elements for any -th order of elliptic and parabolic
Vieta's formulas
In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (more commonly referred to by the Latinised form of his
Trigonometric polynomial
In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n taking on the values of
In calculus, Cavalieri's quadrature formula, named for 17th-century Italian mathematician Bonaventura Cavalieri, is the integral and generalizations thereof. This is the definite integral form; the in
Harmonic polynomial
In mathematics, in abstract algebra, a multivariate polynomial p over a field such that the Laplacian of p is zero is termed a harmonic polynomial. The harmonic polynomials form a vector subspace of t
Laguerre polynomials
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: which is a second-order linear differential equation. This equation has nonsing
P-recursive equation
In mathematics a P-recursive equation is a linear equation of sequences where the coefficient sequences can be represented as polynomials. P-recursive equations are linear recurrence equations (or lin
Invariant polynomial
In mathematics, an invariant polynomial is a polynomial that is invariant under a group acting on a vector space . Therefore, is a -invariant polynomial if for all and . Cases of particular importance
Boas–Buck polynomials
In mathematics, Boas–Buck polynomials are sequences of polynomials defined from analytic functions and by generating functions of the form . The case , sometimes called generalized Appell polynomials,
Jacobian conjecture
In mathematics, the Jacobian 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 has Jaco
Derivation of the Routh array
The Routh array is a tabular method permitting one to establish the stability of a system using only the coefficients of the characteristic polynomial. Central to the field of control systems design,
Ruffini's rule
In mathematics, 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 1804. The rule is a special c
Factorization of polynomials
In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible
Shapiro polynomials
In mathematics, the Shapiro polynomials are a sequence of polynomials which were first studied by Harold S. Shapiro in 1951 when considering the magnitude of specific trigonometric sums. In signal pro
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as a, b a
Delta operator
In mathematics, a delta operator is a shift-equivariant linear operator on the vector space of polynomials in a variable over a field that reduces degrees by one. To say that is shift-equivariant mean
Denisyuk polynomials
In mathematics, Denisyuk polynomials Den(x) or Mn(x) are generalizations of the Laguerre polynomials introduced by given by the generating function .
Theory of equations
In algebra, the theory of equations is the study of algebraic equations (also called "polynomial equations"), which are equations defined by a polynomial. The main problem of the theory of equations w
HOMFLY polynomial
In the mathematical field of knot 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 the
Hiptmair–Xu preconditioner
In mathematics, Hiptmair–Xu (HX) preconditioners are preconditioners for solving and problems based on the auxiliary space preconditioning framework. An important ingredient in the derivation of HX pr
Principal root of unity
In mathematics, a principal n-th root of unity (where n is a positive integer) of a ring is an element satisfying the equations In an integral domain, every primitive n-th root of unity is also a prin
Angelescu polynomials
In mathematics, the Angelescu polynomials πn(x) are a series of polynomials generalizing the Laguerre polynomials introduced by . The polynomials can be given by the generating function , p.41) They c
In mathematics, the stability radius of an object (system, function, matrix, parameter) at a given nominal point is the radius of the largest ball, centered at the nominal point, all of whose elements
Bracket polynomial
In the 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 an invariant of knots or links (as i
Polynomial evaluation
In mathematics and computer science, polynomial evaluation refers to computation of the value of a polynomial when its indeterminates are substituted for some values. In other words, evaluating the po
Polynomial solutions of P-recursive equations
In mathematics a P-recursive equation can be solved for polynomial solutions. Sergei A. Abramov in 1989 and Marko Petkovšek in 1992 described an algorithm which finds all polynomial solutions of those
Division polynomials
In mathematics the division polynomials provide a way to calculate multiples of points on elliptic curves and to study the fields generated by torsion points. They play a central role in the study of
Newton polynomial
In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points. The Newton polynomial is som
Resultant
In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extensi
Coefficient diagram method
In control theory, the coefficient diagram method (CDM) is an algebraic approach applied to a polynomial loop in the parameter space, where a special diagram called a "coefficient diagram" is used as
Wu's method of characteristic set
Wenjun Wu's method is an algorithm for solving multivariate polynomial equations introduced in the late 1970s by the Chinese mathematician Wen-Tsun Wu. This method is based on the mathematical concept
Bombieri norm
In mathematics, the Bombieri norm, named after Enrico Bombieri, is a norm on homogeneous polynomials with coefficient in or (there is also a version for non homogeneous univariate polynomials). This n
Multiplicative sequence
In mathematics, a multiplicative sequence or m-sequence is a sequence of polynomials associated with a formal group structure. They have application in the cobordism ring in algebraic topology.
Stable polynomial
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 roots lie in the open left half-plane, or *
Bernoulli polynomials
In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–Mac
Graph polynomial
In mathematics, a graph polynomial is a graph invariant whose values are polynomials. Invariants of this type are studied in algebraic graph theory.Important graph polynomials include: * The characte
Bell polynomials
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 and Bell numbers. They also occur in many
Divided power structure
In mathematics, specifically commutative algebra, a divided power structure is a way of making expressions of the form meaningful even when it is not possible to actually divide by .
Algebraic function
In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms,
Constant term
In mathematics, a constant term is a term in an algebraic expression that does not contain any variables and therefore is constant. For example, in the quadratic polynomial the 3 is a constant term. A
Bernoulli umbra
In Umbral calculus, Bernoulli umbra is an , a formal symbol, defined by the relation , where is the index-lowering operator, also known as evaluation operator and are Bernoulli numbers, called moments
Square-free polynomial
In mathematics, a square-free polynomial is a polynomial defined over a field (or more generally, an integral domain) that does not have as a divisor any square of a non-constant polynomial. A univari
Geometrical properties of polynomial roots
In mathematics, a univariate polynomial of degree n with real or complex coefficients has n complex roots, if counted with their multiplicities. They form a multiset of n points in the complex plane.
In mathematics, Padovan polynomials are a generalization of Padovan sequence numbers. These polynomials are defined by: The first few Padovan polynomials are: The Padovan numbers are recovered by eval
Lehmer's conjecture
Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer. The conjecture asserts that there is an absolute constant such tha
Cubic equation
In algebra, a cubic equation in one variable is an equation of the form in which a is nonzero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of th
Mittag-Leffler polynomials
In mathematics, the Mittag-Leffler polynomials are the polynomials gn(x) or Mn(x) studied by Mittag-Leffler. Mn(x) is a special case of the Meixner polynomial Mn(x;b,c) at b = 0, c = -1.
Regular chain
In computer algebra, a regular chain is a particular kind of triangular set in a multivariate polynomial ring over a field. It enhances the notion of characteristic set.
Polynomials calculating sums of powers of arithmetic progressions
The polynomials calculating sums of powers of arithmetic progressions are polynomials in a variable that depend both on the particular arithmetic progression constituting the basis of the summed power
Abel polynomials
The Abel polynomials in mathematics form a polynomial sequence, the nth term of which is of the form The sequence is named after Niels Henrik Abel (1802-1829), the Norwegian mathematician. This polyno
Q-difference polynomial
In combinatorial mathematics, the q-difference polynomials or q-harmonic polynomials are a polynomial sequence defined in terms of the q-derivative. They are a generalized type of Brenke polynomial, a
Littlewood polynomial
In mathematics, a Littlewood polynomial is a polynomial all of whose coefficients are +1 or −1.Littlewood's problem asks how large the values of such a polynomial must be on the unit circle in the com
Primitive part and content
In algebra, the content of a polynomial with integer coefficients (or, more generally, with coefficients in a unique factorization domain) is the greatest common divisor of its coefficients. The primi
External ray
An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set.Although this curve is only rarely a half-line (ray) it is called a ray because it is an image of a ray. External ra