Are Buchberger’s criteria necessary for the chain condition?

Buchberger’s Gröbner basis theory plays a fundamental role in symbolic computation. The resulting algorithms essentially carry out several S-polynomial reductions. In his Ph.D. thesis and later publication Buchberger showed that sometimes one can skip S-polynomial reductions if the leading terms of polynomials satisfy certain criteria. A question naturally arises: Are Buchberger’s criteria also necessary for skipping S-polynomial reductions? In this paper, after making the question more precise (in terms of a chain condition), we show the answer to be “almost, but not quite”: necessary when there are four or more polynomials, but not necessary when there are exactly three polynomials. For that case, we found an extension to Buchberger’s criteria that is necessary as well as sufficient.