  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.
