Ore algebra explained

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

Notes and References

1. Chyzak . Frédéric . Salvy . Bruno . 1998 . Non-commutative Elimination in Ore Algebras Proves Multivariate Identities . Journal of Symbolic Computation . 26 . 2 . 187–227 . Elsevier . 10.1006/jsco.1998.0207 .