On Catalan trees and the Jacobian conjecture
type: Item
instance of: scholarly article
title: On Catalan trees and the Jacobian conjecture
zbMATH Open document ID: 0955.05116
author: Dan Singer
published in: The Electronic Journal of Combinatorics
publication date: 2001-01-09T00:00:00Z
full work available at URL: https://eudml.org/doc/120787
full work available at URL: v8i1r2.html
review text: 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 \(G=(x_1-H_1,\dots,x_n-H_n)\).
Mathematics Subject Classification ID: 05E99
Mathematics Subject Classification ID: 05C05
Mathematics Subject Classification ID: 14R15
Mathematics Subject Classification ID: 05A99
zbMATH DE Number: 1549893
zbMATH Keywords: formal tree expansions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1549893