Gr?bner bases for syzygy modules of border bases

摘要

Given an order ideal O and an O-border basis of a 0-dimensional polynomial ideal, it was shown by Huibregtse that the liftings of the neighbor syzygies (i.e. of the fundamental syzygies of neighboring border terms) form a system of generators for the syzygy module of the border basis. We elaborate on Huibregtse's proof and transform it into explicit algorithmic form. Based on this, we are able to exhibit explicit conditions on a module term ordering τ such that the liftings of the neighbor syzygies are in fact a τ-Gr?bner basis. Finally, we construct term orderings satisfying these conditions in an explicit algorithmic way.