Construction of Janet Bases II. Polynomial Bases

Involutive Bases in the Weyl Algebra

“…Computational experiments by Gerdt et al (2001) have shown that in the polynomial case it is often better to use an extension of the "monomial" completion algorithm of Figure 2. It is shown in Figure 3 and uses two simple subalgorithms the details of which we omit, as they are obvious: InvAutoReduce L, (F ) involutively autoreduces the given finite set F ; InvNormalForm L, (g, F ) computes the involutive normal form of g with respect to the set F .…”

“…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.…”

On a method for finding the roots of an ideal

“…If the Janet basis [3][4][5] of the ideal is constructed, there is no need for finding the elements of Ω separately, because they are already available in the structure of the Janet tree [3,4]. Indeed, this set coincides…”

“…with the set of monomials in the non-leaf nodes of the Janet tree [3,4]. This is proved in the following theorem [6].…”

Structure Analysis of Polynomial Modules with Pommaret Bases

“…A very efficient C program for the construction of Janet bases is described together with benchmarks in [4]. We have provided a generic implementation for determining involutive bases (with respect to arbitrary involutive divisions) for ideals in polynomial algebras of solvable type within the computer algebra system MuPAD (see www.mupad.de).…”