On the Relation Between Pommaret and Janet Bases
Vladimir P. Gerdt
TL;DR
The paper investigates the relationship between Pommaret and Janet bases within involutive bases, establishing that if an ideal has a finite Pommaret basis, it is precisely its minimal Janet basis. It introduces an improved JanetBasis algorithm that either computes the finite Pommaret basis (when it exists) or yields a Janet basis (not necessarily minimal), and it generalizes these results to linear differential ideals. The work shows that the finiteness of a Pommaret basis and the minimality of the corresponding Janet basis are tightly linked via PJ-autoreduction, enabling practical computation with provable termination due to noetherity. This contributes a unified, algorithmic framework for polynomial and differential ideals, with implications for Gröbner-like computations and the analysis of $\delta$-singularities in differential equations.
Abstract
In this paper the relation between Pommaret and Janet bases of polynomial ideals is studied. It is proved that if an ideal has a finite Pommaret basis then the latter is a minimal Janet basis. An improved version of the related algorithm for computation of Janet bases, initially designed by Zharkov, is described. For an ideal with a finite Pommaret basis, the algorithm computes this basis. Otherwise, the algorithm computes a Janet basis which need not be minimal. The obtained results are generalized to linear differential ideals.