A note on multidegrees of automorphisms of the form (expD)⋆

Abstract

Let k be a field of characteristic zero. For any polynomial mapping F=(F1,...,Fn):kn→kn by multidegree of F we mean the followingn-tuple of natural numbers mdegF=(degF1,...,degFn). Let us denote by k[x]=k[x1,...,xn] a ring of polynomials innvariables x1,...,xn over k. If D:k[x]→k[x] is a locally nilpotent k-derivation, then one can define the automorphisme xpD of k-algebra k[x] and then the polynomial automorphism (expD)⋆ of kn. In this note we present a general upper bound of mdeg(expD)⋆ in the case of a triangular derivation D, and also show that this estimation is exact.