Ideal-specific elimination orders form a star-shaped region

TL;DR

This paper introduces a new geometric perspective on Gr"obner fans, showing that ideal-specific elimination orders form a star-shaped, non-convex region, enabling earlier termination of elimination algorithms.

Contribution

It provides an improved stopping criterion for Gr"obner walks and reveals that the set of ideal-specific elimination cones forms a star-shaped, non-convex region.

Findings

Elimination cones can be in the interior of the positive orthant.

The cones form a star-shaped region, not convex.

Earlier termination of Gr"obner walks is possible.

Abstract

This paper shows that Gr\"obner walks aiming for the elimination of variables from a polynomial ideal can be terminated much earlier than previously known. To this end we provide an improved stopping criterion for a known Gr\"obner walk algorithm for the elemination of variables. This results from two new geometric insights on Gr\"obner fans: We show that for any given ideal I \subset K[x_1, ..., x_n] the collection of Gr\"obner cones corresponding to I-specific elimination orders may contain Gr\"obner cones in the relative interior of the positive orthant. Moreover we prove that the corresponding Gr\"obner cones form a star-shaped region (the center being the set of all universal elimination vectors) which contrary to first intuition in general is not convex.