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

The structure of the F5 algorithm to compute Gröbner bases makes it very efficient. However, it is not clear whether it terminates for all inputs, not even for "regular sequences". 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 in dissertations, and ensure termination by checking for a Gröbner 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.