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


Kreuzer M., Sipal B., Le Ngoc Long L. N. L.

BEITRAGE ZUR ALGEBRA UND GEOMETRIE-CONTRIBUTIONS TO ALGEBRA AND GEOMETRY, vol.61, no.3, pp.515-532, 2020 (ESCI) identifier identifier

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.