Random planar trees and the Jacobian conjecture

TL;DR

This paper introduces a probabilistic approach to the Jacobian conjecture using random tree models and Markov chains, providing new insights and an approximate solution to a longstanding mathematical problem.

Contribution

It develops a novel probabilistic framework involving multi-type branching processes and subtree shuffling in Catalan trees to analyze the Jacobian conjecture.

Findings

Probabilistic formulation of the Jacobian conjecture via rooted trees

Construction of a Markov chain on large Catalan trees with uniform stationary distribution

An asymptotic proof that inverses of Keller maps have small high-degree coefficients

Abstract

We develop a probabilistic approach to the celebrated Jacobian conjecture, which states that any Keller map (i.e. any polynomial mapping $F\colon \mathbb{C}^n \to \mathbb{C}^n$ whose Jacobian determinant is a nonzero constant) has a compositional inverse which is also a polynomial. The Jacobian conjecture may be formulated in terms of a problem involving labellings of rooted trees; we give a new probabilistic derivation of this formulation using multi-type branching processes. Thereafter, we develop a simple and novel approach to the Jacobian conjecture in terms of a problem involving shuffling subtrees of $d$-Catalan trees, i.e. planar $d$-ary trees. We also show that, if one can construct a certain Markov chain on large $d$-Catalan trees which updates its value by randomly shuffling certain nearby subtrees, and in such a way that the stationary distribution of this chain is uniform, then the Jacobian conjecture is true. Finally, we use the local limit theory of large random trees to show that the subtree shuffling conjecture is true in a certain asymptotic sense, and thereafter use our machinery to prove an approximate version of the Jacobian conjecture, stating that inverses of Keller maps have small power series coefficients for their high degree terms.