Difference algebra
Introduction '''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  together with a ring endomorphism . Often it is assumed that  is injective. When  is a field one speaks of a ''difference field''. A classical example of a difference field is the field  of rational functions with the difference operator  given by . 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 . A ''difference algebra'' over a difference field  is a difference ring  with a -algebra structure such that  is a morphism of difference rings, i.e. extends . A difference algebra that is a field is called a ''difference field extension''.
Algebraic difference equations The difference polynomial ring  over a difference field  in the (difference) variables  is the polynomial ring over  in the infinitely many variables . It becomes a difference algebra over  by extending  from  to  as suggested by the naming of the variables.By a ''system of algebraic difference'' equations over  one means any subset  of . If  is a difference algebra over  the solutions of  in  are:Classically one is mainly interested in solutions in difference field extensions of . For example, if  and  is the field of meromorphic functions on  with difference operator  given by , then the fact that the gamma function  satisfies the functional equation  can be restated abstractly as .
Difference varieties 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 . 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  for some .There is a one-to-one correspondence between the difference varieties defined by algebraic difference equations in the variables  and certain ideals in , namely the perfect difference ideals of . One of the basic theorems in difference algebra asserts that every ascending chain of perfect difference ideals in  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.
See also * Finite difference* Recurrence relation* Functional equation* Differential algebra
Notes
References *Alexander Levin (2008),  Difference algebra, Springer, *Richard M. Cohn (1979),  Difference algebra, R.E. Krieger Pub. Co.,
External links ** The  home page of Zoé Chatzidakis has several online surveys discussing (the model theory of) difference fields.