Characterizations of border bases
Border bases are a system of generators of some zero-dimensional ideals which behave numerically better than Gröbner bases. The usual presentations and characterizations of border bases differ from those of Gröbner bases. In this paper border bases are presented and characterized as Gröbner bases. There is a border division algorithm in which degree is replaced by border index. Then the border bases are characterized via the special generation property. Normal forms and reduction process are also developed (however reduction is not Noetherian). In order to introduce a Buchberger criterion the authors characterize the border bases in terms of the commutation of the multiplication matrices (the ideals are zero dimensional); then they define neighborhood between border terms and these are used to prove the Buchberger criterion for border bases. Finally the paper covers the computation of syzygies; the authors compute syzygies for border terms and these are lifted to border bases; once again the neighbor borders make the computation of syzygies easier.
scientific article; zbMATH DE number 2153395