On Exponentiality of Automorphisms of Aⁿ of Order p in Characteristic $$p>0$$

Let X be an integral affine scheme of characteristic $$p>0$$ p > 0 , and σ σ a non-identity automorphism of X . If σ σ is exponential , i.e., induced from a Gₐ G a -action on X , then σ σ is obviously of order p . It is easy to see that the converse is not true in general. In fact, there exists X which admits an automorphism of order p , but admits no non-trivial Gₐ G a -actions. However, the situation is not clear in the case where X is the affine space ARⁿ A R n , because ARⁿ A R n admits various Gₐ G a -actions as well as automorphisms of order p . In this paper, we study exponentiality of automorphisms of ARⁿ A R n of order p , where the difficulty stems from the non-uniqueness of Gₐ G a -actions inducing an exponential automorphism. Our main results are as follows. (1) We show that the triangular automorphisms of ARⁿ A R n of order p are exponential in some low-dimensional cases. (2) We construct a non-exponential automorphism of ARⁿ A R n of order p for each n≥ 2 n ≥ 2 . Here, R is any UFD which is not a field. (3) We investigate the Gₐ G a