Hilbert schemes of zero-dimensional ideals in a polynomial ring can be covered with suitable affine open subschemes whose construction is achieved using border bases. Moreover, border bases have proved to be an excellent tool for describing zero-dimensional ideals when the coefficients are inexact. And in this situation they show a clear advantage with respect to Gröbner bases which, neverthele...

Border bases have recently attracted a lot of attention. Here we study the problem of generalizing one of the main tools of Gröbner 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 sch...

Gröbner basis detection (GBD) is defined as follows: given a set of polynomials, decide whether there exists and if “yes” find a term order such that the set of polynomials is a Gröbner basis. This problem was proposed by Gritzmann and Sturmfels (1993) and it was shown to be NP-hard by Sturmfels and Wiegelmann. We investigate the computational complexity of this problem when the given set of po...

Border bases, a generalization of Gröbner bases, have actively been researched during recent years due to their applicability to industrial problems. In [6] Kehrein and Kreuzer formulated the so called Border Basis Algorithm, an algorithm which allows the computation of border bases that relate to a degree compatible term ordering. In this paper we extend the original Border Basis Algorithm in ...

We extend the theory and the algorithms of Border Bases to systems of Laurent polynomial equations, defining “toric” roots. Instead of introducing new variables and new relations to saturate by the variable inverses, we propose a more efficient approach which works directly with the variables and their inverse. We show that the commutation relations and the inversion relations characterize tori...