A Combinatorial Approach to Involution and Delta-Regularity I: Involutive Bases in Polynomial Algebras of Solvable Type
Reference: Applicable Algebra in Engineering, Communication and Computing, 20 (2009) 207-259

This first part discusses the basic theory of involutive bases. I treat at once the theory within a rather general class of non-commutative algebras: the polynomial algebras of solvable type (note that I generalise the original definition given by Kandry-Rodi and Weispfenning following Kredel). This class includes besides the ordinary commutative polynomials e.g. rings of linear differential or difference operators, universal enveloping algebras and some quantum algebras. In the case of an algebra over a coefficient ring, the question whether or not the algebra is Noetherian becomes non-trivial and I review three different approaches to proving a basis theorem. As we must distinguish between left and right ideals in a non-commutative setting, I also study the construction of involutive bases for two-sided ideals.

Two further new aspects are involutive bases with respect to arbitrary semigroup orders, as they appear in local computations, and involutive bases in polynomial algebras over rings. In the former case, I analyse besides Lazard's approach (treated for the special case of the Weyl algebra already in [19] ) also the approach via Mora's normal form. In the latter case, some assumptions must be made on the commutation relations in the polynomial algebra for the involutive completion algorithm to terminate. In both cases, in general only weak involutive bases exist. This new concept is introduced here for the first time: such weak bases are still Gröbner bases but they do not define a disjoint decomposition of the ideal.

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