Difference algebra is a branch of mathematics concerned with the study of difference (or functional) equations from the algebraic point of view.
Difference algebra is analogous to differential algebra but concerned with difference equations rather than differential equations.
As an independent subject it was initiated by Joseph Ritt and his student Richard Cohn.
together with a ring endomorphism
of rational functions with the difference operator
The role of difference rings in difference algebra is similar to the role of commutative rings in commutative algebra and algebraic geometry.
A morphism of difference rings is a morphism of rings that commutes with
is a morphism of difference rings, i.e.
A difference algebra that is a field is called a difference field extension.
The difference polynomial ring
as suggested by the naming of the variables.
By a system of algebraic difference equations over
are Classically one is mainly interested in solutions in difference field extensions of
is the field of meromorphic functions on
, then the fact that the gamma function
satisfies the functional equation
Intuitively, a difference variety over a difference field
is the set of solutions of a system of algebraic difference equations over
This definition has to be made more precise by specifying where one is looking for the solutions.
Usually one is looking for solutions in the so-called universal family of difference field extensions of
[1][2] Alternatively, one may define a difference variety as a functor from the category of difference field extensions of
to the category of sets, which is of the form
There is a one-to-one correspondence between the difference varieties defined by algebraic difference equations in the variables
, namely the perfect difference ideals of
[3] One of the basic theorems in difference algebra asserts that every ascending chain of perfect difference ideals in
This result can be seen as a difference analog of Hilbert's basis theorem.
Difference algebra is related to many other mathematical areas, such as discrete dynamical systems, combinatorics, number theory, or model theory.
While some real life problems, such as population dynamics, can be modeled by algebraic difference equations, difference algebra also has applications in pure mathematics.
For example, there is a proof of the Manin–Mumford conjecture using methods of difference algebra.
[4] The model theory of difference fields has been studied.