A combinatorial approach to involution and δ-regularity I: involutive bases in polynomial algebras of solvable type

Involutive bases are a special form of non-reduced Gröbner bases with additional combinatorial properties. Their origin lies in the Janet–Riquier theory of linear systems of partial differential equations. We study them for a rather general class of polynomial algebras including also non-commutative algebras like those generated by linear differential and difference operators or universal enveloping algebras of (finite-dimensional) Lie algebras. We review their basic properties using the novel concept of a weak involutive basis and present concrete algorithms for their construction. As new original results, we develop a theory for involutive bases with respect to semigroup orders (as they appear in local computations) and over coefficient rings, respectively. In both cases it turns out that generally only weak involutive bases exist.