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 Plus annual

Presents the access of premium content as premium feature

Premium Content

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 Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The famous F5 algorithm for computing Gr\"obner basis was presented byFaug\`ere in 2002 without complete proofs for its correctness. The currentauthors have simplified the original F5 algorithm into an F5 algorithm inBuchberger's style (F5B algorithm), which is equivalent to original F5algorithm and may deduce some F5-like versions. In this paper, the F5Balgorithm is briefly revisited and a new complete proof for the correctness ofF5B algorithm is proposed. This new proof is not limited to homogeneous systemsand does not depend on the strategy of selecting critical pairs (i.e. thestrategy deciding which critical pair is computed first) such that any strategycould be utilized in F5B (F5) algorithm. From this new proof, we find that thespecial reduction procedure (F5-reduction) is the key of F5 algorithm, somaintaining this special reduction, various variation algorithms becomeavailable. A natural variation of F5 algorithm, which transforms original F5algorithm to a non-incremental algorithm, is presented and proved in this paperas well. This natural variation has been implemented over the Boolean ring. Thetwo revised criteria in this natural variation are also able to reject almostall unnecessary computations and few polynomials reduce to 0 in most examples.

A New Proof for the Correctness of F5 (F5-Like) Algorithm