Predicting zero reductions in Gr\"obner basis computations

Since Buchberger's initial algorithm for computing Gr\"obner 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 M\"oller. A relatively new approach are signature-based criteria which were first used in Faug\`ere'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 Gr\"obner bases and their computations might be achieved.Comment: 25 pages, 3 figure

Mimesis and Metaphor

"Representation", "imitation" and "mirroring" have proved to be insufficient translations of the concept of mimesis. Walter Benjamin's notion of "mimetic potential" offers a different view on the qualities of mimesis. Benjamin stresses the importance of language and its mediality to mimesis; for him language is the "höchste Stufe des mimetischen Verhaltens und das vollkommenste Archiv der unsinnlichen Ähnlichkeit" (Benjamin 1991, Bd. II/1, 213) He considers mimesis on the level of linguistic mediality. In the following I will try to outline the mimetic potential of metaphors in literary texts which focus on their linguistic mediality. As Paul Ricoeur suggests, "the possibility that metaphorical discourse says something about reality collides with the apparent constitution of poetic discourse, which seems to be essentially non-referential and centred on itself. To this non-referential conception of poetic discourse I oppose the idea that the suspension of literal reference is the condition for the release of a power of second-degree reference which is properly poetic reference. Thus, to use an expression borrowed from Jakobson, one must not speak only of split sense but of 'split reference' as well." (Ricoeur 1977, 6