On Gr\"obner Basis Detection for Zero-dimensional Ideals

TL;DR

This paper investigates the computational complexity of Gr"obner basis detection for zero-dimensional ideals, proving NP-hardness and proposing a polynomial-time algorithm for fixed numbers of variables.

Contribution

It extends NP-hardness results to zero-dimensional ideals and introduces an efficient algorithm for cases with a fixed number of variables.

Findings

GBD for zero-dimensional ideals is NP-hard.

Proposed polynomial-time algorithm for fixed variable count.

Extends complexity understanding of Gr"obner basis detection.

Abstract

The Gr\"obner basis detection (GBD) is defined as follows: Given a set of polynomials, decide whether there exists -and if "yes" find- a term order such that the set of polynomials is a Gr\"obner basis. This problem was shown to be NP-hard by Sturmfels and Wiegelmann. We show that GBD when studied in the context of zero dimensional ideals is also NP-hard. An algorithm to solve GBD for zero dimensional ideals is also proposed which runs in polynomial time if the number of indeterminates is a constant.