It is well known that Border bases generalize Gröbner bases. From this point of view, it is natural to ask oneself if a theorem in Gröbner basis theory has an equivalent in Border basis theory. In this paper, the problem of generalizing one of the main tools of Gröbner basis theory, namely the flat deformation to the leading term ideal, to the Border basis setting is studied. To do this, the authors introduce border basis schemes and universal border basis families, with these two concepts they achieve to reformulate the initial problem as the search for a certain rational curve on a border basis scheme. Then, the system of generators of the vanishing ideal of the border basis scheme is constructed and the question of how to minimalize it is studied. Finally, the previous situation for the particular case of homogeneous ideals is treated.
Reviewer: Manuel Avelino Insua Hermo (Vigo)