Random planar trees and the Jacobian conjecture

Abstract

We develop a probabilistic approach to the celebrated Jacobian conjecture. This more than 80 year old problem from algebraic geometry states that any Keller map (i.e. any polynomial mapping F : C n → 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 branching processes. Thereafter, we develop a simple and novel approach to the Jacobian conjecture in terms of a conjecture about shuffling subtrees of planar rooted d-ary trees. We show that such a shuffling conjecture is true in a certain asymptotic sense. We next use this 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.