2007/10/14 by Martin Kreuzer, Lorenzo Robbiano, Kreuzer, Martin +1
Engineering · Materials Science · #13P10 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #Elasticity and Wave Propagation #FOS: Mathematics #Material Properties and Applications #Structural mechanics and materials
paper · pdf · doi:10.48550/arxiv.0710.2641
openalex publication_date 2007/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.