Dynamical Gröbner bases
Since its introduction, the notion of Gröbner basis has been generalized in many ways. In [\textit{I. Yengui}, ``Computing a Gröbner basis of a polynomial ideal over a principal domain'', preprint (2004)], the author generalized Gröbner bases to discrete valuation domains, giving a simple decision procedure for the ideal membership problem in this case. The first part of the current paper is a direct generalization of that work to noetherian valuation rings. As a sample application, the author computes two examples of the structure of codes over two different finite-chain rings. A third example shows that in the integral case, the totally ordered group corresponding to the valuation must be archimedian in order to guarantee that Buchberger's Algorithm will terminate in a finite number of steps. The author then generalizes Gröbner bases to principal rings by computing a ``dynamical Gröbner basis'', i.e. a tree of Gröbner bases in localizations of the original ring at finitely generated multiplicative subsets. The author provides a detailed example of the construction, along with an example of how the ideal membership problem can be answered with the resulting basis. While the dynamical method avoids the ``expensive problem of factorizing an element in a principal domain into a finite product of irreducible elements'', it introduces its own computational problems in the form of redundant leaves ``due to the fact that one can obtain the same leaf in different ways''. The author does mention several shortcuts that may help to alleviate this problem.
scientific article; zbMATH DE number 5078115