The Jacobian Conjecture, a Reduction of the Degree to the Quadratic Case

Abstract

The Jacobian Conjecture states that any locally invertible polynomial system in Cn is globally invertible with polynomial inverse. Bass et al. (Bull Am Math Soc 7(2):287–330, 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.