On Catalan Trees and the Jacobian Conjecture

Abstract

New combinatorial properties of Catalan trees are established and used to prove a number of algebraic results related to the Jacobian conjecture. Let $F=(x_1+H_1,x_2+H_2,\dots,x_n+H_n)$ be a system of $n$ polynomials in $C[x_1,x_2,\dots,x_n]$, the ring of polynomials in the variables $x_1,x_2, \dots, x_n$ over the field of complex numbers. Let $H=(H_1,H_2,\dots,H_n)$. Our principal algebraic result is that if the Jacobian of $F$ is equal to 1, the polynomials $H_i$ are each homogeneous of total degree 2, and $({{\partial H_i}\over {\partial x_j}})^3=0$, then $H\circ H\circ H=0$ and $F$ has an inverse of the form $G=(G_1,G_2,\dots,G_n)$, where each $G_i$ is a polynomial of total degree $\le6$. We prove this by showing that the sum of weights of Catalan trees over certain equivalence classes is equal to zero. We also show that if all of the polynomials $H_i$ are homogeneous of the same total degree $d\ge2$ and $({{\partial H_i}\over {\partial x_j}})^2=0$, then $H\circ H=0$ and the inverse of $F$ is $G=(x_1-H_1,\dots,x_n-H_n)$.