In this paper, we have made the attempt to classify the integrand basis of
all two-loop diagrams in pure four-dimension space-time. Our classification
includes the topology of two-loop diagrams which determines the structure of
denominators, and the set of numerators under different kinematic
configurations of external momenta by using Gr\"{o}bner basis method. In our
study, the variety defined by setting all propagators to on-shell has played an
important role. We discuss the structure of variety and how it splits to
various irreducible branches when external momenta at each corner of diagrams
satisfy some special kinematic conditions. This information is crucial to the
numerical or analytical fitting of coefficients for integrand basis in
reduction process.Comment: 52 pages, 9 figures. v2 reference added, v3 published versio
