On the Jung-van der Kulk decomposition into Pascal finite factors

Abstract

Combining the Jung--van der Kulk theorem with the conjugacy invariance of the Pascal finite class, we show that every polynomial automorphism $F$ of the plane over an arbitrary field $K$, satisfying $F(0) = 0$, decomposes into the form $F = \diag(\det J_F, 1) \circ P_1 \circ \dots \circ P_s$, where all $P_i$ are Pascal finite automorphisms. Since every Pascal finite automorphism has Jacobian determinant equal to 1, the diagonal factor is the only obstacle: $F$ is a composition of Pascal finite maps if and only if $\det J_F = 1$. In particular, Question~3.1 from \cite{ABCH2} has a positive answer in dimension 2 in any characteristic, which constitutes an analogue of the Exponential Generators Conjecture in positive characteristic. In characteristic $p$, the factors can be chosen to have an order dividing $p^2$.