Border bases and order ideals: a polyhedral characterization

Details Border bases and order ideals: a polyhedral characterization Border bases and order ideals: a polyhedral characterization

2009-12-08

arxiv.org/abs/0912.1502v2

Mathematics

Commutative Algebra

14 pages; Corrected typos and an example

Scientific paper

Border bases arise as a canonical generalization of Gr\"obner bases. We extend our previous work [arXiv:0911.0859] to arbitrary order ideals: we devise a polyhedral characterization of order ideals and border bases: order ideals that support a border basis correspond one-to-one to integral points of the order ideal polytope. In particular, we establish a crucial connection between the ideal and its combinatorial structure. Based on this characterization we adapt the classical border basis algorithm to allow for computing border bases for arbitrary order ideals, which are independent of term orderings. We also show that finding a preference-optimal order ideal that supports a border basis is NP-hard.

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