Border basis detection is NP-complete

TL;DR

This paper proves that determining whether a given set of generators forms a border basis of an ideal with respect to some order ideal is an NP-complete problem, highlighting its computational difficulty.

Contribution

It establishes that border basis detection (BBD) is NP-complete, extending the understanding of the computational complexity of basis detection problems.

Findings

BBD is NP-complete.

The complexity of BBD parallels that of GBD.

Highlights computational challenges in algebraic basis detection.

Abstract

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 Gr\"obner bases termed as Gr\"obner 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.