The Hilbert zonotope and a polynomial time algorithm for universal Gröbner bases

Authors:
Eric Babson;Shmuel Onn;Rekha Thomas
Affiliations:
University of Washington at Seattle, Seattle, WA 98195, USA;Technion, Israel Institute of Technology, 32000 Haifa, Israel and University of California at Davis, Davis, CA 95616, USA;University of Washington at Seattle, Seattle, WA 98195, USA
Venue:
Advances in Applied Mathematics
Year:
2003

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Abstract

We provide a polynomial time algorithm for computing the universal Grobner basis of any polynomial ideal having a finite set of common zeros in fixed number of variables. One ingredient of our algorithm is an effective construction of the state polyhedron of any member of the Hilbert scheme Hilb^d"n of n-long d-variate ideals, enabled by introducing the Hilbert zonotopeH^d"n and showing that it simultaneously refines all state polyhedra of ideals on Hilb^d"n.