In
mathematics, an algebraic differential equation is a
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
that can be expressed by means of
differential algebra
In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A ...
. There are several such notions, according to the concept of differential algebra used.
The intention is to include equations formed by means of
differential operators, in which the coefficients are
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s of the variables (e.g. the
hypergeometric equation). Algebraic differential equations are widely used in
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 software for manipulating mathematical expression ...
and
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
.
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.
Formulations
*
Derivations ''D'' can be used as algebraic analogues of the formal part of
differential calculus, so that algebraic differential equations make sense in
commutative rings.
*The theory of
differential fields was set up to express
differential Galois theory 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
In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The mo ...
is something that transcribes easily into
algebraic geometry, giving an algebraic analogue of the way
systems of differential equations are geometrically represented by
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to ev ...
s with connections.
*The concept of
jet
Jet, Jets, or The Jet(s) may refer to:
Aerospace
* Jet aircraft, an aircraft propelled by jet engines
** Jet airliner
** Jet engine
** Jet fuel
* Jet Airways, an Indian airline
* Wind Jet (ICAO: JET), an Italian airline
* Journey to Enceladus a ...
can be described in purely algebraic terms, as was done in part of
Grothendieck's
EGA
Ega or EGA may refer to:
Military
* East German Army, the common western name for the National People's Army
* Eagle, Globe, and Anchor, the emblem of the United States Marine Corps
People
* Aega (mayor of the palace), 7th-century noble of Neus ...
project.
*The theory of
D-module
In 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. Since around 1970, ''D''-module theory has be ...
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 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, involving only the algebraic operations additi ...
: solving equations typically produces novel
transcendental function
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function.
In other words, a transcendental function "transcends" algebra in that it cannot be expressed alg ...
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
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ve ...
s and linear equations). This case stands in relation with the whole theory roughly as
invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...
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
*
*{{SpringerEOM, title=Extension of a differential field , id=Extension_of_a_differential_field , oldid=18279 , first=A.V. , last=Mikhalev , first2=E.V. , last2=Pankrat'ev
Differential equations
Differential algebra