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, 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 n ...

. 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 functions of the variables (e.g. the hypergeometric equation). Algebraic differential equations are widely used in computer algebra 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, "Mat ...

. 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, so that algebraic differential equations make sense in commutative rings. *The theory of differential fields 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 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 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 every p ...

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-modules 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 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 ver ...

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