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

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 $$F(0) = 0$$ F ( 0 ) = 0 , decomposes into the form F = \,diag\,( JF, 1) ∘ P₁ ∘ ∘ Pₛ F = diag ( det J F , 1 ) ∘ P 1 ∘ ⋯ ∘ P s , where all Pᵢ P i are Pascal finite automorphisms. Since every Pascal finite automorphism has Jacobian determinant equal to 1, the diagonal factor is the only obstacle, and F is a composition of Pascal finite maps if and only if JF = 1 det J F = 1 . In particular, Question 3.1 from [2] has a positive answer in dimension 2 in any characteristic, which constitutes an analogue of the exponential generators theorem in positive characteristic. In characteristic p , the factors can be chosen to have an order dividing p² p 2 .