Deformations of Border Bases

TL;DR

This paper explores the extension of flat deformation techniques from Groebner bases to border bases, introducing schemes and methods to construct and analyze these deformations.

Contribution

It develops border basis schemes and universal families, providing new tools and explicit constructions for deformations in border basis theory.

Findings

Border basis schemes can be constructed via different generator systems.

Homogeneous border basis schemes are affine spaces under certain conditions.

Explicit deformations are achievable for homogeneous ideals in specific cases.

Abstract

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.