Automatic differentiation
In mathematics and computer algebra, automatic differentiation (AD), also called algorithmic differentiation, computational differentiation, auto-differentiation, or simply autodiff, is a set of techn
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
Dimension of an algebraic variety
In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways. Some of these definitions are of geometric nature, while some ot
Elementary function
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rati
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.
Janet basis
In mathematics, a Janet basis is a normal form for systems of linear homogeneous partial differential equations (PDEs) that removes the inherent arbitrariness of any such system. It was introduced in
Journal of Symbolic Computation
The Journal of Symbolic Computation is a peer-reviewed monthly scientific journal covering all aspects of symbolic computation published by Academic Press and then by Elsevier. It is targeted to both
Elimination theory
In commutative algebra and algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating some variables between polynomials of several variables, in order to s
Gröbner fan
In computer algebra, the Gröbner fan of an ideal in the ring of polynomials is a concept in the theory of Gröbner bases. It is defined to be a fan consisting of cones that correspond to different mono
Abramov's algorithm
In mathematics, particularly in computer algebra, Abramov's algorithm computes all rational solutions of a linear recurrence equation with polynomial coefficients. The algorithm was published by Serge
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
Conway polynomial (finite fields)
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 a standard representation of Fpn as a splittin
Cantor–Zassenhaus algorithm
In computational algebra, the Cantor–Zassenhaus algorithm is a method for factoring polynomials over finite fields (also called Galois fields). The algorithm consists mainly of exponentiation and poly
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
Binary expression tree
A binary expression tree is a specific kind of a binary tree used to represent expressions. Two common types of expressions that a binary expression tree can represent are algebraic and boolean. These
Symbolic integration
In calculus, symbolic integration is the problem of finding a formula for the antiderivative, or indefinite integral, of a given function f(x), i.e. to find a differentiable function F(x) such that Th
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
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
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
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
Gosper's algorithm
In mathematics, Gosper's algorithm, due to Bill Gosper, is a procedure for finding sums of hypergeometric terms that are themselves hypergeometric terms. That is: suppose one has a(1) + ... + a(n) = S
Symbolic-numeric computation
In mathematics and computer science, symbolic-numeric computation is the use of software that combines symbolic and numeric methods to solve problems.
Liouvillian function
In mathematics, the Liouvillian functions comprise a set of functions including the elementary functions and their repeated integrals. Liouvillian functions can be recursively defined as integrals of
Risch algorithm
In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert He
Berlekamp's algorithm
In mathematics, particularly computational algebra, Berlekamp's algorithm is a well-known method for factoring polynomials over finite fields (also known as Galois fields). The algorithm consists main
Pollard's kangaroo algorithm
In computational number theory and computational algebra, Pollard's kangaroo algorithm (also Pollard's lambda algorithm, see below) is an algorithm for solving the discrete logarithm problem. The algo
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
Schwartz–Zippel lemma
In mathematics, the Schwartz–Zippel lemma (also called the DeMillo-Lipton-Schwartz–Zippel lemma) is a tool commonly used in probabilistic polynomial identity testing, i.e. in the problem of determinin
Bareiss algorithm
In mathematics, the Bareiss algorithm, named after , is an algorithm to calculate the determinant or the echelon form of a matrix with integer entries using only integer arithmetic; any divisions that
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
Buchberger's algorithm
In the theory of multivariate polynomials, Buchberger's algorithm is a method for transforming a given set of polynomials into a Gröbner basis, which is another set of polynomials that have the same c
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
Sum of radicals
In computational complexity theory, there is an open problem of whether some information about a sum of radicals may be computed in polynomial time depending on the input size, i.e., in the number of
Symbolic regression
Symbolic regression (SR) is a type of regression analysis that searches the space of mathematical expressions to find the model that best fits a given dataset, both in terms of accuracy and simplicity
Sturm's theorem
In mathematics, the Sturm sequence of a 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 ex
Berlekamp–Zassenhaus algorithm
In mathematics, in particular in computational algebra, the Berlekamp–Zassenhaus algorithm is an algorithm for factoring polynomials over the integers, named after Elwyn Berlekamp and Hans Zassenhaus.
Computer algebra
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and softwa
Faugère's F4 and F5 algorithms
In computer algebra, the Faugère F4 algorithm, by Jean-Charles Faugère, computes the Gröbner basis of an ideal of a multivariate polynomial ring. The algorithm uses the same mathematical principles as
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
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.
Gröbner basis
In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal i
Kronecker substitution
Kronecker substitution is a technique named after Leopold Kronecker for determining the coefficients of an unknown polynomial by evaluating it at a single value. If p(x) is a polynomial with integer c
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
Ore algebra
In computer algebra, an Ore algebra is a special kind of iterated Ore extension that can be used to represent linear functional operators, including linear differential and/or recurrence operators. Th