The Jacobian conjecture

The Jacobian conjecture involves the map \(y= x - V(x)\) where \(y, x\) are n-dimensional vectors, \(V(x)\) is a symmetric polynomial of degree \(d\) for which the Jacobian hypothesis holds: \( e^{Tr \ln(1- V'(x))} =1,\ \forall x\). The conjecture states that the inverse map (\(x\) as a function of \(y\)) is also polynomial. The proof is inspired by perturbative field theory. We express the inverse map \(F(y)= y+ V(F(y))\) as a perturbative expansion which is a sum of partially ordered connected trees. We use the property : \(\frac{d F_{k}}{dy_{k}}= (\frac{1}{1-V'(F)})_{k,k} =1+ \sum _{q\ge 1} \frac{1}{q} (Tr (V'(F))^{q})_{with\ q\ edges\ of\ index\ k}\) to extract inductively in the index \(k\) all the sub traces in the expansion of the inverse map. We obtain \(F= F(|\le n)\ \ e^{- Tr \ln(1-V'(F(|\le n)))}\) By the Jacobian hypothesis \(e^{- Tr \ln(1-V'(F(|\le n)))} =1\) and a straightforward graphical argument gives that \(degree \ in\ y \ of \ F(|\le n)\le d^{2^{n} -2}\)