|Title||Characterizations of border bases|
|Author(s)|| Achim Kehrein, Martin Kreuzer|
|Type||Article in Journal|
|Abstract||This paper presents characterizations of border bases of zero-dimensional|
polynomial ideals that are analogous to the known characterizations of
Gröbner bases. Based on a Border Division Algorithm, a variant of the
usual Division Algorithm, we characterize border bases as border prebases
with one of the following equivalent properties: special generation,
generation of the border form ideal, confluence of the corresponding
rewrite relation, reduction of S-polynomials to zero, and lifting of syzygies.
The last characterization relies on a detailed study of the relative position
of the border terms and their syzygy module. In particular, a border prebasis
is a border basis if and only if all fundamental syzygies of the
neighboring border terms lift; these liftings are easy to compute.
|Keywords||border basis, Gröbner basis, S-polynomial, lifting of syzygies, border term|
|Journal||Journal of Pure and Applied Algebra|