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.
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.
The Weyl algebraW of differential operators with polynomial coefficients can be considered; certain modulesM can be used to express differential equations, according to the presentation of M.
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: solving equations typically produces novel transcendental functions. 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 surfaces 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.
