  In this paper we investigate the Schwinger parametric representation for the
Feynman amplitudes of the recently discovered renormalizable $\phi^4_4$ quantum
field theory on the Moyal non commutative ${\mathbb R^4}$ space. This
representation involves new {\it hyperbolic} polynomials which are the
non-commutative analogs of the usual "Kirchoff" or "Symanzik" polynomials of
commutative field theory, but contain richer topological information.
