  It is well known that the integral identity conjecture is of prime importance
in Kontsevich-Soibelman's theory of motivic Donaldson-Thomas invariants for
non-commutative Calabi-Yau threfolds. In this article we consider its numerical
version and make it a complete demonstration in the case where the potential is
a polynomial and the ground field is algebraically closed. The foundamental
tool is the Berkovich spaces whose crucial point is how to use the comparison
theorem for nearby cycles as well as the K\"{u}nneth isomorphism for cohomology
with compact support.
