The Geometry of Ideal-specific Elimination-Orders and Improvements to Elimination

Details The Geometry of Ideal-specific Elimination-Orders and Improvements to Elimination The Geometry of Ideal-specific Elimination-Orders and Improvements to Elimination

2012-02-13

arxiv.org/abs/1202.2727v2

Mathematics

Algebraic Geometry

19 pages, 5 figures

Scientific paper

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.

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