Ideal-specific elimination orders form a star-shaped region

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.