mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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, an ...

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 natur ...

. 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 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 that accepts a function and return ...

s, 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 rat ...

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 expressions ...

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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

. 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

Derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...

s ''D'' can be used as algebraic analogues of the formal part of

differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...

, so that algebraic differential equations make sense in

commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...

s. *The theory of

differential field 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 natu ...

s was set up to express

differential Galois theory In mathematics, differential Galois theory studies the Galois groups of differential equations. Overview Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential field ...

in algebraic terms. *The

Weyl algebra In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form : f_m(X) \partial_X^m + f_(X) \partial_X^ + \cdots + f_1(X) \partial_X + f_0(X). More prec ...

''W'' of differential operators with polynomial coefficients can be considered; certain

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...

s ''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 geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, 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 every po ...

s with connections. *The concept of jet can be described in purely algebraic terms, as was done in part of Grothendieck's EGA project. *The theory of

D-module In mathematics, a ''D''-module is a module (mathematics), module over a ring (mathematics), ring ''D'' of differential operators. The major interest of such ''D''-modules is as an approach to the theory of linear partial differential equations. Sin ...

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 additio ...

: 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 alge ...

s. The case of algebraic solutions is however of considerable interest; the classical

Schwarz list In the mathematical theory of special functions, Schwarz's list or the Schwartz table is the list of 15 cases found by when hypergeometric functions can be expressed algebraically. More precisely, it is a listing of parameters determining the ca ...

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 In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ...

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 vers ...

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 In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to re ...

. 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