Basics

• Polynomial
  Polynomial
  In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
  
  
• Coefficient
  Coefficient
  In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
  
  
• Monomial
  Monomial
  In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...
  
  
• Polynomial long division
  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 generalised version of the familiar arithmetic technique called long division...
  
  
• Synthetic division
  Synthetic division
  Synthetic division is a method of performing polynomial long division, with less writing and fewer calculations. It is mostly taught for division by binomials of the formx - a,\ but the method generalizes to division by any monic polynomial...
  
  
• Polynomial factorization
  Polynomial factorization
  In mathematics and computer algebra, polynomial factorization refers to factoring a polynomial into irreducible polynomials over a given field.-Formulation of the question:...
  
  
• Rational function
  Rational function
  In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
  
  
• Partial fraction
  Partial fraction
  In algebra, the partial fraction decomposition or partial fraction expansion is a procedure used to reduce the degree of either the numerator or the denominator of a rational function ....
  
  
  • Partial fraction decomposition over R
• Vieta's formulas
• Integer-valued polynomial
  Integer-valued polynomial
  In mathematics, an integer-valued polynomial P is a polynomial taking an integer value P for every integer n. Certainly every polynomial with integer coefficients is integer-valued. There are simple examples to show that the converse is not true: for example the polynomialgiving the triangle...
  
  
• Algebraic equation
• Factor theorem
  Factor theorem
  In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem.The factor theorem states that a polynomial f has a factor if and only if f=0....
  
  
• Polynomial remainder theorem
  Polynomial remainder theorem
  In algebra, the polynomial remainder theorem or little Bézout's theorem is an application of polynomial long division. It states that the remainder of a polynomial f\, divided by a linear divisor x-a\, is equal to f \,.- Example :...
  
  

Elementary abstract algebra

See also Theory of equations below.

• Polynomial ring
  Polynomial ring
  In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
  
  
• Greatest common divisior of two polynomials
• Symmetric function
  Symmetric function
  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...
  
  
• Homogeneous polynomial
  Homogeneous polynomial
  In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...
  
  
• Polynomial SOS
  Polynomial SOS
  In mathematics, a form h of degree 2m in the real n-dimensional vector x is sum of squares of forms if and only if there exist forms g_1,\ldots,g_k of degree m such that...
  
   (sum of squares)

Theory of equations

Theory of equations

In mathematics, the theory of equations comprises a major part of traditional algebra. Topics include polynomials, algebraic equations, separation of roots including Sturm's theorem, approximation of roots, and the application of matrices and determinants to the solving of equations.From the point...

• Binomial theorem
  Binomial theorem
  In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...
  
  
• Blossom (functional)
  Blossom (functional)
  In numerical analysis, a blossom is a functional that can be applied to any polynomial, but is mostly used for Bézier and spline curves and surfaces....
  
  
• Root of a function
• Nth root
  Nth root
  In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...
  
   (Radical)
  • Surd
    Surd
    Surd may be:* A voiceless consonant* An Nth root, any mathematical expression such as a square root, cube root or higher root* Surd, Hungary, a village in Zala county, Hungary...
    
    
  • Square root
    Square root
    In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
    
    
  • Methods of computing square roots
    Methods of computing square roots
    There are several methods for calculating the principal square root of a nonnegative real number. For the square roots of a negative or complex number, see below.- Rough estimation :...
    
    
  • Cube root
  • Root of unity
    Root of unity
    In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
    
    
  • Constructible number
    Constructible number
    A point in the Euclidean plane is a constructible point if, given a fixed coordinate system , the point can be constructed with unruled straightedge and compass...
    
    
  • Complex conjugate root theorem
    Complex conjugate root theorem
    In 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 a and b real numbers, then its complex conjugate a − bi is also a root of P.It follows from this , that if the degree...
    
    
• Algebraic element
  Algebraic element
  In mathematics, if L is a field extension of K, then an element a of L is called an algebraic element over K, or just algebraic over K, if there exists some non-zero polynomial g with coefficients in K such that g=0...
  
  
• Horner scheme
  Horner scheme
  In numerical analysis, the Horner scheme , named after William George Horner, is an algorithm for the efficient evaluation of polynomials in monomial form. Horner's method describes a manual process by which one may approximate the roots of a polynomial equation...
  
  
• Rational root theorem
• Gauss's lemma (polynomial)
  Gauss's lemma (polynomial)
  In algebra, in the theory of polynomials , Gauss's lemma is either of two related statements about polynomials with integer coefficients:...
  
  
• Irreducible polynomial
  Irreducible polynomial
  In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....
  
  
  • Eisenstein's criterion
    Eisenstein's criterion
    In mathematics, Eisenstein's criterion gives an easily checked sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers...
    
    
  • Primitive polynomial
    Primitive polynomial
    In field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite extension field GF...
    
    
• Quadratic equation
  Quadratic equation
  In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...
  
  
  • Completing the square
    Completing the square
    In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the formax^2 + bx + c\,\!to the formIn this context, "constant" means not depending on x. The expression inside the parenthesis is of the form ...
    
    
• Cubic function
  Cubic function
  In mathematics, a cubic function is a function of the formf=ax^3+bx^2+cx+d,\,where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function...
  
  
• Quartic function
  Quartic function
  In mathematics, a quartic function, or equation of the fourth degree, is a function of the formf=ax^4+bx^3+cx^2+dx+e \,where a is nonzero; or in other words, a polynomial of degree four...
  
  
• Quintic equation
  Quintic equation
  In mathematics, a quintic function is a function of the formg=ax^5+bx^4+cx^3+dx^2+ex+f,\,where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero...
  
  
  • Abel–Ruffini theorem
    Abel–Ruffini theorem
    In algebra, the Abel–Ruffini theorem states that there is no general algebraic solution—that is, solution in radicals— to polynomial equations of degree five or higher.- Interpretation :...
    
    
  • Quintic function
  • Bring radical
    Bring radical
    In algebra, a Bring radical or ultraradical of a complex number a is a root of the polynomialx^5+x+a. \,In algebra, a Bring radical or ultraradical of a complex number a is a root of the polynomialx^5+x+a. \,In algebra, a Bring radical or ultraradical of a complex number a is a root...
    
    
• Fundamental theorem of algebra
  Fundamental theorem of algebra
  The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
  
  
• Hurwitz polynomial
  Hurwitz polynomial
  In mathematics, a Hurwitz polynomial, named after Adolf Hurwitz, is a polynomial whose coefficients are positive real numbers and whose zeros are located in the left half-plane of the complex plane, that is, the real part of every zero is negative...
  
  
• Tschirnhaus transformation
  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. It may be defined conveniently by means of field theory, as the transformation on minimal polynomials implied by a...
  
  
• Galois theory
  Galois theory
  In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
  
  
• Discriminant of a polynomial
  • Resultant
• Elimination theory
  Elimination theory
  In commutative algebra and algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating between polynomials of several variables....
  
  
  • Gröbner basis
    Gröbner basis
    In computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating subset of an ideal I in a polynomial ring R...
    
    
  • Regular chain
    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.- Introduction :...
    
    
  • Triangular decomposition
    Triangular decomposition
    In computer algebra, a triangular decomposition of a polynomial system S is a set of simpler polynomial systems S_1,\ldots, S_e such that a point is a solution of S if and only if it is a solution of one of the systems S_1,\ldots, S_e....
    
    
• Sturm's theorem
  Sturm's theorem
  In mathematics, Sturm's theorem is a symbolic procedure to determine the number of distinct real roots of a polynomial. It was named for Jacques Charles François Sturm...
  
  
• Descartes' rule of signs
  Descartes' rule of signs
  In mathematics, Descartes' rule of signs, first described by René Descartes in his work La Géométrie, is a technique for determining the number of positive or negative real roots of a polynomial....
  
  

Polynomial interpolation

Polynomial interpolation

In numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial: given some points, find a polynomial which goes exactly through these points.- Applications :...

• Lagrange polynomial
  Lagrange polynomial
  In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of distinct points x_j and numbers y_j, the Lagrange polynomial is the polynomial of the least degree that at each point x_j assumes the corresponding value y_j...
  
  
• Runge's phenomenon
  Runge's phenomenon
  In the mathematical field of numerical analysis, Runge's phenomenon is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree...
  
  
• Spline (mathematics)
  Spline (mathematics)
  In mathematics, a spline is a sufficiently smooth piecewise-polynomial function. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low-degree polynomials, while avoiding Runge's phenomenon for higher...
  
  

Linear algebra

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

• Characteristic polynomial
  Characteristic polynomial
  In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....
  
  
• Minimal polynomial
• Invariant polynomial
  Invariant polynomial
  In mathematics, an invariant polynomial is a polynomial P that is invariant under a group \Gamma acting on a vector space V. Therefore P is a \Gamma-invariant polynomial ifP = Pfor all \gamma \in \Gamma and x \in V....
  
  

Named polynomials and polynomial sequence

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...

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• Additive polynomial
  Additive polynomial
  In mathematics, the additive polynomials are an important topic in classical algebraic number theory.-Definition:Let k be a field of characteristic p, with p a prime number. A polynomial P with coefficients in k is called an additive polynomial, or a Frobenius polynomial, ifP=P+P\,as polynomials...
  
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• Appell sequence
• Askey–Wilson polynomials
  Askey–Wilson polynomials
  In mathematics, the Askey–Wilson polynomials are a family of orthogonal polynomials introduced by as q-analogs of the Wilson polynomials. They include many of the other orthogonal polynomials in 1 variable as special or limiting cases, described in the Askey scheme...
  
  
• Bell polynomials
• Bernoulli polynomials
  Bernoulli polynomials
  In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part because they are an Appell sequence, i.e. a Sheffer sequence for the ordinary derivative operator...
  
  
• Bessel polynomials
  Bessel polynomials
  In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions...
  
  
• Binomial type
• Caloric polynomial
  Caloric polynomial
  In differential equations, the mth-degree caloric polynomial is a "parabolically m-homogeneous" polynomial Pm that satisfies the heat equation"Parabolically m-homogeneous" means...
  
  
• Charlier polynomials
  Charlier polynomials
  In mathematics, Charlier polynomials are a family of orthogonal polynomials introduced by Carl Charlier....
  
  
• Chebyshev polynomials
  Chebyshev polynomials
  In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...
  
  
• Ehrhart polynomial
  Ehrhart polynomial
  In mathematics, an integral polytope has an associated Ehrhart polynomial which encodes the relationship between the volume of a polytope and the number of integer points the polytope contains...
  
  
• Exponential polynomial
  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.-In fields:...
  
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• Favard's theorem
  Favard's theorem
  In mathematics, Favard's theorem, also called the Shohat–Favard theorem, states that a sequence of polynomials satisfying a suitable 3-term recurrence relation is a sequence of orthogonal polynomials...
  
  
• Fibonacci polynomials
• Hahn polynomials
  Hahn polynomials
  In mathematics, the Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Chebyshev in 1875 and rediscovered by . The Hahn class is a name for special cases of Hahn polynomials, including Hahn polynomials, Meixner...
  
  
• Heat polynomial — see caloric polynomial
  Caloric polynomial
  In differential equations, the mth-degree caloric polynomial is a "parabolically m-homogeneous" polynomial Pm that satisfies the heat equation"Parabolically m-homogeneous" means...
  
  
• Heckman–Opdam polynomials
  Heckman–Opdam polynomials
  In mathematics, Heckman–Opdam polynomials Pλ are orthogonal polynomials in several variables associated to root systems...
  
  
• Hermite polynomials
  Hermite polynomials
  In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; and in physics, where...
  
  
• Kravchuk polynomials
  Kravchuk polynomials
  Kravchuk polynomials or Krawtchouk polynomials are discrete orthogonal polynomials associated with the binomial distribution, introduced by .The first few polynomials are:...
  
  
• Laguerre polynomials
  Laguerre polynomials
  In mathematics, the Laguerre polynomials, named after Edmond Laguerre ,are the canonical solutions of Laguerre's equation:x\,y + \,y' + n\,y = 0\,which is a second-order linear differential equation....
  
  
• Laurent polynomial
• Legendre polynomials
  • Spherical harmonic
    Spherical Harmonic
    Spherical Harmonic is a science fiction novel from the Saga of the Skolian Empire by Catherine Asaro. It tells the story of Dyhianna Selei , the Ruby Pharaoh of the Skolian Imperialate, as she strives to reform her government and reunite her family in the aftermath of a devastating interstellar...
    
    
• Macdonald polynomials
• Meixner polynomials
• Newton polynomial
  Newton polynomial
  In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form...
  
  
• Orthogonal polynomials
  Orthogonal polynomials
  In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...
  
  
• Orthogonal polynomials on the unit circle
  Orthogonal polynomials on the unit circle
  In mathematics, orthogonal polynomials on the unit circle are families of polynomials that are orthogonal with respect to integration over the unit circle in the complex plane, for some probability measure on the unit circle. They were introduced by ....
  
  
• Racah polynomials
  Racah polynomials
  In mathematics, Racah polynomials are orthogonal polynomials named after Giulio Racah, as their orthogonality relations are equivalent to his orthogonality relations for Racah coefficients....
  
  
• Rook polynomial
  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 the same row or column...
  
  
• Schur polynomial
  Schur polynomial
  In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of...
  
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• Sheffer sequence
• Touchard polynomials
• Wilkinson's polynomial
• Wilson polynomials
  Wilson polynomials
  In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials....
  
  

Algorithm

Algorithm

In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

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• Karatsuba multiplication
• Lenstra–Lenstra–Lovász lattice basis reduction algorithm
  Lenstra–Lenstra–Lovász lattice basis reduction algorithm
  The LLL-reduction algorithm is a polynomial time lattice reduction algorithm invented by Arjen Lenstra, Hendrik Lenstra and László Lovász in 1982, see...
  
   (for polynomial factorization)
• Schönhage–Strassen algorithm