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 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. A difference ring is a commutative ring R together with a ring endomorphism \sigma\colon R\to R . Often it is assumed that \sigma is injective. When R is a field one speaks of a difference field. A classical example of a difference field is the field K=\mathbb{C}(x) of rational functions with the difference operator \sigma given by \sigma(f(x))=f(x+1) . 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 \sigma . A difference algebra over a difference field K is a difference ring R with a K -algebra structure such that K\to R is a morphism of difference rings, i.e. \sigma\colon R\to R extends \sigma\colon K\to K . A difference algebra that is a field is called a difference field extension. The difference polynomial ring K\{y\}=K\{y_1,\ldots,y_n\} over a difference field K in the (difference) variables y_1,\ldots,y_n is the polynomial ring over K in the infinitely many variables \sigma^i(y_j),\ (i\in\mathbb{N}, 1\leq j\leq n) . It becomes a difference algebra over K by extending \sigma from K to K\{y\} as suggested by the naming of the variables. By a system of algebraic difference equations over K one means any subset F of K\{y\} . If R is a difference algebra over K the solutions of F in R are \mathbb{V}_R(F)=\{a\in R^n|\ f(a)=0 \text{ for all } f\in F\}. Classically one is mainly interested in solutions in difference field extensions of K . For example, if K=\mathbb{C}(x) and R is the field of meromorphic functions on \mathbb{C} with difference operator \sigma given by \sigma(f(x)) = f(x+1) , then the fact that the gamma function \Gamma satisfies the functional equation \Gamma(x+1) = x\Gamma(x) can be restated abstractly as \Gamma\in\mathbb{V}_R(\sigma(y_1)-xy_1) . ==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.

Text derived from https://en.wikipedia.org/w/index.php?oldid=1343900551 under CC BY-SA 4.0; modifications have been made.