  The Jacobian Conjecture states that any locally invertible polynomial system
in C^n is globally invertible with polynomial inverse. C. W. Bass et al. (1982)
proved a reduction theorem stating that the conjecture is true for any degree
of the polynomial system if it is true in degree three. This degree reduction
is obtained with the price of increasing the dimension n. We prove here a
theorem concerning partial elimination of variables, which implies a reduction
of the generic case to the quadratic one. The price to pay is the introduction
of a supplementary parameter 0<n'< n, parameter which represents the dimension
of a linear subspace where some particular conditions on the system must hold.
We first give a purely algebraic proof of this reduction result and we then
expose a distinct proof, in a Quantum Field Theoretical formulation, using the
intermediate field method.
