Ore algebra

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In computer algebra, an Ore algebra is a special kind of iterated Ore extension that can be used to represent linear functional operators, including linear differential and/or recurrence operators.[1] The concept is named after Øystein Ore.

Definition

Let K {\displaystyle K} {\displaystyle K} be a (commutative) field and A = K [ x 1 , … , x s ] {\displaystyle A=K[x_{1},\ldots ,x_{s}]} {\displaystyle A=K[x_{1},\ldots ,x_{s}]} be a commutative polynomial ring (with A = K {\displaystyle A=K} {\displaystyle A=K} when s = 0 {\displaystyle s=0} {\displaystyle s=0}). The iterated skew polynomial ring A [ ∂ 1 ; σ 1 , δ 1 ] ⋯ [ ∂ r ; σ r , δ r ] {\displaystyle A[\partial _{1};\sigma _{1},\delta _{1}]\cdots [\partial _{r};\sigma _{r},\delta _{r}]} {\displaystyle A[\partial _{1};\sigma _{1},\delta _{1}]\cdots [\partial _{r};\sigma _{r},\delta _{r}]} is called an Ore algebra when the σ i {\displaystyle \sigma _{i}} {\displaystyle \sigma _{i}} and δ j {\displaystyle \delta _{j}} {\displaystyle \delta _{j}} commute for i ≠ j {\displaystyle i\neq j} {\displaystyle i\neq j}, and satisfy σ i ( ∂ j ) = ∂ j {\displaystyle \sigma _{i}(\partial _{j})=\partial _{j}} {\displaystyle \sigma _{i}(\partial _{j})=\partial _{j}}, δ i ( ∂ j ) = 0 {\displaystyle \delta _{i}(\partial _{j})=0} {\displaystyle \delta _{i}(\partial _{j})=0} for i > j {\displaystyle i>j} {\displaystyle i>j}.

Properties

Ore algebras satisfy the Ore condition, and thus can be embedded in a (skew) field of fractions.

The constraint of commutation in the definition makes Ore algebras have a non-commutative generalization theory of Gröbner basis for their left ideals.

References

  1. Chyzak, Frédéric; Salvy, Bruno (1998). "Non-commutative Elimination in Ore Algebras Proves Multivariate Identities" (PDF). Journal of Symbolic Computation. 26 (2). Elsevier: 187–227. doi:10.1006/jsco.1998.0207.