Difference algebra

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.
Difference rings, difference fields and difference algebras.
A "difference ring" is a commutative ring formula_1 together with a ring endomorphism formula_2. Often it is assumed that formula_3 is injective. When formula_4 is a field one speaks of a "difference field". A classical example of a difference field is the field formula_5 of rational functions with the difference operator formula_3 given by formula_7. 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 formula_3. A "difference algebra" over a difference field formula_9 is a difference ring formula_4 with a formula_9-algebra structure such that formula_12 is a morphism of difference rings, i.e. formula_2 extends formula_14. A difference algebra that is a field is called a "difference field extension".
Algebraic difference equations.
The difference polynomial ring formula_15 over a difference field formula_9 in the (difference) variables formula_17 is the polynomial ring over formula_9 in the infinitely many variables formula_19. It becomes a difference algebra over formula_9 by extending formula_3 from formula_9 to formula_23 as suggested by the naming of the variables.
By a "system of algebraic difference" equations over formula_9 one means any subset formula_25 of formula_23. If formula_4 is a difference algebra over formula_9 the solutions of formula_25 in formula_4 are
Classically one is mainly interested in solutions in difference field extensions of formula_9. For example, if formula_5 and formula_4 is the field of meromorphic functions on formula_35 with difference operator formula_3 given by formula_7, then the fact that the gamma function formula_38 satisfies the functional equation formula_39 can be restated abstractly as formula_40.
Difference varieties.
Intuitively, a "difference variety" over a difference field formula_9 is the set of solutions of a system of algebraic difference equations over formula_9. 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 formula_9. Alternatively, one may define a difference variety as a functor from the category of difference field extensions of formula_9 to the category of sets, which is of the form formula_45 for some formula_46.
There is a one-to-one correspondence between the difference varieties defined by algebraic difference equations in the variables formula_17 and certain ideals in formula_23, namely the perfect difference ideals of formula_23. One of the basic theorems in difference algebra asserts that every ascending chain of perfect difference ideals in formula_23 is finite. This result can be seen as a difference analog of Hilbert's basis theorem.
Applications.
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. The model theory of difference fields has been studied.