Modifying Faug\`ere's F5 Algorithm to ensure termination

The structure of the F5 algorithm to compute Gr\"obner bases makes it very efficient. However, while it is believed to terminate for so-called regular sequences, it is not clear whether it terminates for all inputs. This paper has two major parts. In the first part, we describe in detail the difficulties related to a proof of termination. In the second part, we explore three variants that ensure termination. Two of these have appeared previously only in dissertations, and ensure termination by checking for a Gr\"obner basis using traditional criteria. The third variant, F5+, identifies a degree bound using a distinction between "necessary" and "redundant" critical pairs that follows from the analysis in the first part. Experimental evidence suggests this third approach is the most efficient of the three.