The Galois theory of linear differential equations is the analogue of the classical Galois theory of polynomial equations for linear differential equations. The natural analogue of the field in the classical case is the differential field. This is a field equipped with a derivation. We have constructed a basic facility for differential fields and rings. These types can be built from the algebraic function field or affine algebra types. Our medium term goal is to construct a fast solver for linear differential equations.
Construction of the rational differential field and the more general differential ring
Coercions, arithmetic and functionality for elements as for the underlying ring.
Changing the derivation of a differential ring.
Extending the constant ring of a differential ring
Wronskian matrix and Wronskian determinant
The differential constant field of a rational differential field
Ring and field extensions of differential rings and fields
Construction of a differential ideal
Quotient rings, rings and field of fractions of differential rings and fields
Creation of a differential operator ring
Coercion, arithmetic and simple predicates for elements
Accessing coefficients of elements
Changing the derivation of a differential operator ring.
Changing the operator ring by extending the constant ring
Making a differential operator monic.
Adjoint of an operator
Applying an operator to an element of its basering
Euclidean algorithms, left and right (extended) GCD, (extended) left LCM
Companion matrix of an operator.
Determination of whether a place is regular, regular singular or irregular singular at an operator
All singular points of an operator
The indicial polynomial of an operator at a place
All rational solutions of an operator within a rational differential field
Newton polygon and Newton polynomial
Differential field extension of the base ring of an operator by adjoining a formal solution and formal derivatives
The symmetric power of an operator