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Random planar trees and the Jacobian conjecture
We develop a probabilistic approach to the celebrated Jacobian conjecture,
which states that any Keller map (i.e. any polynomial mapping 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 about shuffling
subtrees of -Catalan trees, i.e. planar -ary trees. We also show that, if
one can construct a certain Markov chain on large -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 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.Comment: 36 pages, 4 figures. Section 2.5 added, Section 3 expanded, further
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