Since their introduction for algorithm F5 in 2002, signature Gröbner
bases have brought large improvements to the performances of Gröbner
bases algorithms for polynomial systems over fields. Furthermore, they
contain additional data which can be used, for example, to compute the
module of syzygies of an ideal or to compute coefficients in terms of
the input generators.
In this talk, we present two variants of Buchberger's algorithm
computing signature Gröbner bases over principal ideal domains. The
first one is adapted from Kandri-Rody and Kapur's algorithm (1988),
whereas the second one uses the ideas developed in the algorithms by Pan
(1989) and Lichtblau (2012). The fact that the algorithms have a
structure similar to Buchberger's allows to examine more powerful
signature criteria than in previous signature-based algorithms over
rings, and we will explain how the differences in constructions between
the two algorithms lead to different criteria being applicable.
(Joint work with Maria Francis)