In modeling physical systems it is sometimes useful to construct border bases of 0-dimensional polynomial ideals which are contained in the ideal generated by a given set of polynomials. We define and construct such subideal border bases, provide som
e basic properties and generalize a suitable variant of the Buchberger-Moeller algorithm as well as the AVI-algorithm to the subideal setting. The subideal version of the AVI-algorithm is then applied to an actual industrial problem.
Here we study the problem of generalizing one of the main tools of Groebner basis theory, namely the flat deformation to the leading term ideal, to the border basis setting. After showing that the straightforward approach based on the deformation to
the degree form ideal works only under additional hypotheses, we introduce border basis schemes and universal border basis families. With their help the problem can be rephrased as the search for a certain rational curve on a border basis scheme. We construct the system of generators of the vanishing ideal of the border basis scheme in different ways and study the question of how to minimalize it. For homogeneous ideals, we also introduce a homogeneous border basis scheme and prove that it is an affine space in certain cases. In these cases it is then easy to write down the desired deformations explicitly.
