The Hilbert Zonotope and a Polynomial Time Algorithm for Universal Grobner Bases

We provide a polynomial time algorithm for computing the universal Gr\"obner 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 zonotope H^d_n and showing that it simultaneously refines all state polyhedra of ideals on Hilb^d_n.