Computing Coupled Border Bases

Kehrein and Kreuzer (J Pure Appl Algebra 205(2):279–295, 2006) presented an algorithm, referred to as the BorderBasis algorithm, for computing border bases. The main objective of this paper is to improve this computation by applying new structures from the theory of Grobner bases. In doing so, based on the method developed by Gao et al. (Math Comput 85(297):449–465, 2016), a new variant of the BorderBasis algorithm is proposed to compute simultaneously a border basis and also a Grobner basis for the syzygy module of the input polynomials (this pair of bases will be called a coupled border basis). The new algorithm, and also the original form of the BorderBasis algorithm, have been implemented in the computer algebra systems Maple and ApCoCoA. We compare the performance of our implementation of these algorithms via a set of benchmark polynomials. Our experiments show that our algorithm performs more efficiently than the original BorderBasis algorithm.