English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Signature-based algorithms have become a standard approach for Gr\"obnerbasis computations for polynomial systems over fields, but how to extend thesetechniques to coefficients in general rings is not yet as well understood.  In this paper, we present a proof-of-concept signature-based algorithm forcomputing Gr\"obner bases over commutative integral domains. It is adapted froma general version of M\"oller's algorithm (1988) which considers reductions bymultiple polynomials at each step. This algorithm performs reductions withnon-decreasing signatures, and in particular, signature drops do not occur.When the coefficients are from a principal ideal domain (e.g. the ring ofintegers or the ring of univariate polynomials over a field), we provecorrectness and termination of the algorithm, and we show how to use signatureproperties to implement classic signature-based criteria to eliminate someredundant reductions. In particular, if the input is a regular sequence, thealgorithm operates without any reduction to 0.  We have written a toy implementation of the algorithm in Magma. Earlyexperimental results suggest that the algorithm might even be correct andterminate in a more general setting, for polynomials over a uniquefactorization domain (e.g. the ring of multivariate polynomials over a field ora PID).

A Signature-based Algorithm for computing Computing Gr\"obner Bases over Principal Ideal Domains