On the regularity of the monomial point of a border basis scheme

Abstract

The border basis scheme BO parametrizes all 0-dimensional ideals in K[x1, … , xn] , where K is an arbitrary field, which have a border basis with respect to a given order ideal of terms O. Its vanishing ideal I(BO) is generated by quadratic equations which are easy to describe, and it contains a unique monomial point, namely the point corresponding to the monomial ideal generated by the terms outside O. Based on a detailed study of the generators of I(BO) , we describe a K-basis of the cotangent space m/ m2, where m is the maximal ideal of the monomial point. Moreover, we provide an efficient algorithm to compute such a basis and use it to characterize the regularity of the monomial point. © 2020, The Managing Editors.