Open Access

Elia Bisi, Piotr Dyszewski, Nina Gantert, Samuel G. G. Johnston, Joscha Prochno, Dominik Schmid

• We develop a probabilistic approach to the celebratedJacobian conjecture, which states that any Keller map(i.e. any polynomial mapping F ∶ Cn → Cn whoseJacobian determinant is a non-zero constant) has acompositional inverse which is also a polynomial. TheJacobian conjecture may be formulated in terms of aproblem involving labellings of rooted trees; we give anew probabilistic derivation of this formulation usingmulti-type branching processes. Thereafter, we developa simple and novel approach to the Jacobian conjecturein terms of a problem involving shuffling subtrees ofd-Catalan trees, that is, planar d-ary trees. We alsoshow that, if one can construct a certain Markov chainon large d-Catalan trees which updates its value byrandomly shuffling certain nearby subtrees, and in sucha way that the stationary distribution of this chain is uni-form, then the Jacobian conjecture is true. Finally, weuse the local limit theory of large random trees to showthat the subtree shuffling conjecture isWe develop a probabilistic approach to the celebratedJacobian conjecture, which states that any Keller map(i.e. any polynomial mapping F ∶ Cn → Cn whoseJacobian determinant is a non-zero constant) has acompositional inverse which is also a polynomial. TheJacobian conjecture may be formulated in terms of aproblem involving labellings of rooted trees; we give anew probabilistic derivation of this formulation usingmulti-type branching processes. Thereafter, we developa simple and novel approach to the Jacobian conjecturein terms of a problem involving shuffling subtrees ofd-Catalan trees, that is, planar d-ary trees. We alsoshow that, if one can construct a certain Markov chainon large d-Catalan trees which updates its value byrandomly shuffling certain nearby subtrees, and in sucha way that the stationary distribution of this chain is uni-form, then the Jacobian conjecture is true. Finally, weuse the local limit theory of large random trees to showthat the subtree shuffling conjecture is true in a certainasymptotic sense, and thereafter use our machin-ery to prove an approximate version of the Jacobianconjecture, stating that inverses of Keller maps have small power series coefficients for their high-degreeterms.…