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Predicting zero reductions in Gröbner basis computations

Since Buchberger's initial algorithm for computing Gröbner bases in 1965 many attempts have been taken to detect zero reductions in advance. Buchberger's Product and Chain criteria may be known the most, especially in the installaton of Gebauer and Möller. A relatively new approach are signature-based criteria which were first used in Faugère's F5 algorithm in 2002. For regular input sequences these criteria are known to compute no zero reduction at all. In this paper we give a detailed discussion on zero reductions and the corresponding syzygies. We explain how the different methods to predict them compare to each other and show advantages and drawbacks in theory and practice. With this a new insight into algebraic structures underlying Gröbner bases and their computations might be achieved.