On Catalan Trees and the Jacobian Conjecture

New combinatorial properties of Catalan trees are established and used to prove a number of algebraic results related to the Jacobian conjecture. Let F = (x1 + H1, x2 + H2, . . . , xn + Hn) be a system of n polynomials in C[x1, x2, . . . , xn], the ring of polynomials in the variables x1, x2, . . . , xn over the field of complex numbers. Let H = (H1,H2, . . . ,Hn). Our principal algebraic result is that if the Jacobian of F is equal to 1, the polynomials Hi are each homogeneous of total degree 2, and ( ∂Hi ∂xj ) = 0, then H◦H◦H = 0 and F has an inverse of the form G = (G1, G2, . . . , Gn), where each Gi is a polynomial of total degree ≤ 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 Hi are homogeneous of the same total degree d ≥ 2 and ( ∂Hi ∂xj ) 2 = 0, then H ◦H = 0 and the inverse of F is G = (x1 −H1, . . . , xn −Hn).