Annales Henri Poincaré

, Volume 17, Issue 11, pp 3237–3254 | Cite as

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

  • Axel de GoursacEmail author
  • Andrea Sportiello
  • Adrian Tanasa


The Jacobian Conjecture states that any locally invertible polynomial system in \({{\mathbb{C}}^n}\) 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 \({0 \leq n' \leq n}\), 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.


Partition Function Feynman Rule Polynomial System Outgoing Edge Weyl Algebra 
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© Springer International Publishing 2016

Authors and Affiliations

  • Axel de Goursac
    • 1
    Email author
  • Andrea Sportiello
    • 2
  • Adrian Tanasa
    • 3
    • 4
  1. 1.Chargé de Recherche FNRSUniversité Catholique de LouvainLouvainBelgium
  2. 2.LIPN, Institut Galilée, CNRS UMR 7030, Université Paris 13VilletaneuseFrance
  3. 3.LaBRI, UMR 5800Université de BordeauxTalenceFrance
  4. 4.Horia Hulubei National Institute for Physics and Nuclear EngineeringMagureleRomania

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