On Catalan trees and the Jacobian conjecture
Summary: 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 \(\leq 6\). 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 \(d\geq 2\) and \(({{\partial H_i}\over {\partial x_j}})^2=0\), then \(H\circ H=0\) and the inverse of \(F\) is \(G=(x_1-H_1,\dots,x_n-H_n)\).
scientific article; zbMATH DE number 1549893