Differential Rings (New) [HB 67]

The Galois theory of linear differential equations is the analogue for linear differential equations of the classical Galois theory of polynomial equations. The natural analogue of the field in the classical case is the differential field which is a field equipped with a derivation. We have undertaken to construct a basic facility for differential fields and rings with the medium term goal of constructing a fast solver for linear differential equations.

Differential rings are formed by adding the functionality of a derivative to an ordinary ring in Magma. Additional functionality is available for rational and algebraic function fields. Differential rings can be used to create differential operators and in a wider perspective to consider topics related to differential galois theory.

New Features:

• Differential rings can be created from any ring given a map from the ring to itself and a specified constant field. A special case is that of the rational function field with the usual derivative. These differential rings look like the ring they are created from and inherit all the functionality of that ring and similarly for their elements.

• The ring underlying a differential ring and the derivation can both be retrieved. Differential rings can be tested for equality and a few other properties.

• Arithmetic of elements, as available in the underlying ring, can be performed. In the same way, elements can be tested for equality and for being one or zero. Of course, elements can also be differentiated with respect to the derivation.

• A differential ring can be created from an existing differential ring by changing the derivation or extending the constant field.

• Differential rings can be extended. The permissible extensions include algebraic, exponential, logarithmic and by adjoining the formal solutions of a linear differential operator.

• Ideals and quotients of differential rings can be formed.

• The Wronskian matrix of a sequence of differential ring elements can be computed.

• Rings of differential operators can be created over any differential ring and that differential ring can be retrieved from the operator ring as well as its derivation. Differential operator rings can be compared to determine equality.

• The usual ring element arithmetic can be performed on differential operators. They can be tested for equality and for being one, zero or monic. Elements can be deconstructed into a sequence of terms and the coefficients of monomials can be extracted. The order of a differential operator can be accessed. An monic differential operator can be formed from a given operator as well as an adjoint operator.

• Operators can be applied to elements of the underlying differential field.

• Euclidean left and right division and GCDs and LCMs of operators is available.

• The companion matrix to a differential operator can be created.

• There are functions which examine the interaction between differential operators and places of the underlying differential field.

• Rational solutions of linear differential equations L(y) = 0 and L(y) = g, where g is an element of the underlying differential field of the operator L, can be calculated.

• The Newton polygon of an operator at a place can be created and the newton polynomial of a face on this polygon computed.

• Symmetric powers of operators can be taken and operators can be created with the formal roots of a given polynomial as solutions.