Ideal-specific elimination orders form a star-shaped region

This paper shows that for any given polynomial ideal I⊂K[x1,…,xn]I⊂K[x1,…,xn] the collection of Gröbner cones corresponding to II-specific elimination orders forms a star-shaped region which contrary to first intuition in general is not convex.Moreover we show that the corresponding region may contain Gröbner cones intersecting in the boundary of the Gröbner fan in the origin only. This implies that Gröbner walks aiming for the elimination of variables from a polynomial ideal can be terminated earlier than previously known. We provide a slightly improved stopping criterion for a known Gröbner walk algorithm for the elimination of variables.