Involutive Bases in the Weyl Algebra

Involutive bases are a special kind of non-reduced Gröbner bases. They have been introduced by Gerdt and collaborators for polynomial ideals (see e.g. Zharkov and Blinkov, 1993; Gerdt and Blinkov, 1998a,b) based on ideas from the Janet–Riquier theory of differential equations (Janet, 1929; Riquier, 1910). Involutive bases possess special combinatorial properties: in particular, they define Stanley decompositions (Seiler, 2001). These decompositions were originally introduced for Hilbert function computations (Stanley, 1978), but also proved useful in other applications like invariant theory (Sturmfels and White, 1991; Gatermann, 2000) or the computation of syzygy resolutions (Seiler, 2001). The Janet–Riquier theory also motivated an explicit algorithm for the determination of involutive bases. Experiments by Gerdt et al. (2001) showed that it is fairly efficient and represents a highly competitive alternative to the traditional Buchberger algorithm for the computation of Gröbner bases, even if one is not interested in the additional combinatorial properties of involutive bases. Thus it appears natural to see whether involutive bases can also be introduced for ideals in other than commutative polynomial rings. This paper discusses the case of left ideals in the Weyl algebra§ (Coutinho, 1995). The algorithmic study of D-modules has been pioneered by Briançon and Maisonobe (1984) in the univariate and by Castro-Jiménez (1984, 1987) and Galligo (1985) in the multivariate case. Further references can be found in Saito et al. (2000). For term orders such a generalization is rather straightforward. It was remarked by Apel (1995, 1998) that the theory can be extended to algebras of solvable type which include, in particular, Ore algebras. A more general class of solvable algebras was discussed by Seiler (2001); the special case of linear differential operators was extensively studied by Gerdt (1999).