- Algebraic differential equation
:"Note:

Differential algebraic equation is something different."In

mathematics , an**algebraic differential equation**is adifferential equation that can be expressed by means ofdifferential 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 arerational function s of the variables (e.g. thehypergeometric equation ). Algebraic differential equations are widely used incomputer algebra andnumber theory .A simple concept is that of a

**polynomial vector field**, in other words avector 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 incommutative ring s.

*The theory ofdifferential field s was set up to expressdifferential Galois theory in algebraic terms.

*TheWeyl 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 ofKoszul connection is something that transcribes easily intoalgebraic geometry , giving an algebraic analogue of the way systems of differential equations are geometrically represented byvector bundle s with connections.

*The concept of jet can be described in purely algebraic terms, as was done in part ofGrothendieck 'sEGA project.

*The theory ofD-module s is a global theory of linear differential equations, and has been developed to include substantive results in the algebraic theory (including aRiemann-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 : solving equations typically produces noveltranscendental function s. The case of algebraic solutions is however of considerable interest; the classicalSchwartz 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 finitemonodromy group for the case ofRiemann surface s and linear equations). This case stands in relation with the whole theory roughly asinvariant theory does togroup 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 Mathematics ":

* [*http://eom.springer.de/D/d031830.htm Differential algebra*]

* [*http://eom.springer.de/E/e036960.htm Extension of a differential field*]

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