Algebraic differential equation
Encyclopedia
In mathematics
, an algebraic differential equation is a differential equation
that can be expressed by means of differential algebra
. There are several such notions, according to the concept of differential algebra used.
The intention is to include equations formed by means of differential operator
s, in which the coefficients are rational function
s of the variables (e.g. the hypergeometric equation). Algebraic differential equations are widely used 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 a standard co-ordinate basis as the first partial derivatives with polynomial coefficients. This is a type of first-order algebraic differential operator.
: solving equations typically produces novel transcendental function
s. The case of algebraic solutions is however of considerable interest; the classical Schwarz list deals with the case of the hypergeometric equation. In differential Galois theory the case of algebraic solutions is that in which the differential Galois group G is finite (equivalently, of dimension 0, or of a finite monodromy group for the case of Riemann surface
s and linear equations). This case stands in relation with the whole theory roughly as invariant theory
does to group representation theory. The group G is in general difficult to compute, the understanding of algebraic solutions is an indication of upper bounds for G.
:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an algebraic differential equation is a differential equation
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...
that can be expressed by means of differential algebra
Differential algebra
In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with a derivation, which is a unary function that is linear and satisfies the Leibniz product law...
. There are several such notions, according to the concept of differential algebra used.
The intention is to include equations formed by means of differential operator
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 .This article considers only linear operators,...
s, in which the coefficients are 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:...
s of the variables (e.g. the hypergeometric equation). Algebraic differential equations are widely used in computer algebra and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
.
A simple concept is that of a polynomial vector field, in other words a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
expressed with respect to a standard co-ordinate basis as the first partial derivatives with polynomial coefficients. This is a type of first-order algebraic differential operator.
Formulations
- DerivationDerivation (abstract algebra)In abstract algebra, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field K, a K-derivation is a K-linear map D: A → A that satisfies Leibniz's law: D = b + a.More...
s D can be used as algebraic analogues of the formal part of differential calculusDifferential calculusIn mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
, so that algebraic differential equations make sense in commutative ringCommutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s. - The theory of differential fields was set up to express differential Galois theoryDifferential Galois theoryIn mathematics, differential Galois theory studies the Galois groups of differential equations.Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation, D. Much of...
in algebraic terms. - The Weyl algebra W of differential operators with polynomial coefficients can be considered; certain modules M can be used to express differential equations, according to the presentation of M.
- The concept of Koszul connection is something that transcribes easily into algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, giving an algebraic analogue of the way systems of differential equations are geometrically represented by vector bundleVector bundleIn mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s with connections. - The concept of jetJet (mathematics)In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain...
can be described in purely algebraic terms, as was done in part of Grothendieck's EGAÉléments de géométrie algébriqueThe Éléments de géométrie algébrique by Alexander Grothendieck , or EGA for short, is a rigorous treatise, in French, on algebraic geometry that was published from 1960 through 1967 by the Institut des Hautes Études Scientifiques...
project. - The theory of D-moduleD-moduleIn mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of linear partial differential equations...
s is a global theory of linear differential equations, and has been developed to include substantive results in the algebraic theory (including a Riemann-Hilbert correspondence for higher dimensions).
Algebraic solutions
It is usually not the case that the general solution of an algebraic differential equation is an algebraic functionAlgebraic function
In mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equationwhere the coefficients ai are polynomial...
: solving equations typically produces novel transcendental function
Transcendental function
A transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation...
s. The case of algebraic solutions is however of considerable interest; the classical Schwarz list deals with the case of the hypergeometric equation. In differential Galois theory the case of algebraic solutions is that in which the differential Galois group G is finite (equivalently, of dimension 0, or of a finite monodromy group for the case of Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s and linear equations). This case stands in relation with the whole theory roughly as invariant theory
Invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
does to group representation theory. The group G is in general difficult to compute, the understanding of algebraic solutions is an indication of upper bounds for G.
External links
Links to the Encyclopaedia of MathematicsEncyclopaedia of Mathematics
The Encyclopaedia of Mathematics is a large reference work in mathematics. It is available in book form and on CD-ROM....
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