  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 $degree \ in\ y \ of \ F(|\le
n)\le d^{2^{n} -2}$
