Concepts inComputing GCRDs of approximate differential polynomials
Differentially closed field
In mathematics, a differential field K is differentially closed if every finite system of differential equations with a solution in some differential field extending K already has a solution in K. This concept was introduced by Robinson (1959). Differentially closed fields are the analogues for differential equations of algebraically closed fields for polynomial equations.
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Sylvester matrix
In mathematics, a Sylvester matrix is a matrix associated to two polynomials that provides information about those polynomials. It is named for James Joseph Sylvester.
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Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another (in the style of a higher-order function in computer science). This article considers mainly linear operators, which are the most common type. However, non-linear differential operators, such as the Schwarzian derivative also exist.
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Greatest common divisor
In mathematics, the greatest common divisor (gcd), also known as the greatest common factor (gcf), or highest common factor (hcf), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4. This notion can be extended to polynomials, see greatest common divisor of two polynomials.
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Algebraic structure
In mathematics, and more specifically abstract algebra, the term algebraic structure generally refers to an arbitrary set with one or more finitary operations defined on it. Common examples of structures include groups, rings, fields and lattices. More complex algebraic structures can be defined by introducing multiple operations, different underlying sets, or by altering the defining axioms. Examples of more complex structures include vector spaces, modules and algebras.
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Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication.
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Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines.
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