  In this short review we first recall combinatorial or ($0-$dimensional)
quantum field theory (QFT). We then give the main idea of a standard QFT
method, called the intermediate field method, and we review how to apply this
method to a combinatorial QFT reformulation of the celebrated Jacobian
conjecture on the invertibility of polynomial systems. This approach
establishes a related theorem concerning partial elimination of variables that
implies a reduction of the generic case to the quadratic one. Note that this
does not imply solving the Jacobian conjecture, because one needs to introduce
a supplementary parameter for the dimension of a certain linear subspace where
the system holds.
