3 research outputs found
Comprehensive Border Bases for Zero Dimensional Parametric Polynomial Ideals
In this paper, we extend the idea of comprehensive Gr\"{o}bner bases given by
Weispfenning (1992) to border bases for zero dimensional parametric polynomial
ideals. For this, we introduce a notion of comprehensive border bases and
border system, and prove their existence even in the cases where they do not
correspond to any term order. We further present algorithms to compute
comprehensive border bases and border system. Finally, we study the relation
between comprehensive Gr\"{o}bner bases and comprehensive border bases w.r.t. a
term order and give an algorithm to compute such comprehensive border bases
from comprehensive Gr\"{o}bner bases.Comment: 15 pages, 8 sections and 3 algorithm
Border basis detection is NP-complete
Border basis detection (BBD) is described as follows: given a set of generators of an ideal, decide whether that set of generators is a border basis of the ideal with respect to some order ideal. The motivation for this problem comes from a similar problem related to Grobner bases termed as Grobner basis detection (GBD) which was proposed by Gritzmann and Sturmfels (1993). GBD was shown to be NP-hard by Sturmfels and Wiegelmann (1996). In this paper, we investigate the computational complexity of BBD and show that it is NP-complete