The Jacobian conjecture

Abstract

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: $e^{Tr\ln (1-V^{\prime }(x))}=1,\forall x$. 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 : $\frac{d{F}_k}{d{y}_k}=(\frac{1}{1-V^{\prime }(F)}{)}_{k,k}=1+{\sum }_{q\ge 1}\frac{1}{q}(Tr(V^{\prime }(F){)}^q{)}_{withqedgesofindexk}$ to extract inductively in the index $k$ all the sub traces in the expansion of the inverse map. We obtain $F=F(|\le n)e^{-Tr\ln (1-V^{\prime }(F(|\le n)))}$ By the Jacobian hypothesis $e^{-Tr\ln (1-V^{\prime }(F(|\le n)))}=1$ and a straightforward graphical argument gives that $degreeinyofF(|\le n)\le d^{2^n-2}$