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.

A difference ring is a commutative ring ${\displaystyle R}$ together with a ring endomorphism ${\displaystyle \sigma \colon R\to R}$. Often it is assumed that ${\displaystyle \sigma }$ is injective. When ${\displaystyle R}$ is a field one speaks of a difference field. A classical example of a difference field is the field ${\displaystyle K=\mathbb {C} (x)}$ of rational functions with the difference operator ${\displaystyle \sigma }$ given by ${\displaystyle \sigma (f(x))=f(x+1)}$. 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 ${\displaystyle \sigma }$. A difference algebra over a difference field ${\displaystyle K}$ is a difference ring ${\displaystyle R}$ with a ${\displaystyle K}$-algebra structure such that ${\displaystyle K\to R}$ is a morphism of difference rings, i.e. ${\displaystyle \sigma \colon R\to R}$ extends ${\displaystyle \sigma \colon K\to K}$. A difference algebra which is a field is called a difference field extension.

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