A note on multidegrees of automorphisms of the form \((\exp D)_{\star}\)

Abstract

Let \(k\) be a field of characteristic zero. For any polynomial mapping \(F=(F_1,\ldots,F_n):k^n\rightarrow k^n\) by multidegree of \(F\) we mean the following \(n\)-tuple of natural numbers mdeg \(F=(\deg F_1,\ldots,\deg F_n).\) Let us denote by \(k[x]=k[x_1,\ldots,x_n]\) a ring of polynomials in \(n\) variables \(x_1,\ldots,x_n\) over \(k.\) If \(D:k[x]\rightarrow k[x]\) is a locally nilpotent \(k\)-derivation, then one can define the automorphism \(\exp D\) of \(k\)-algebra \(k[x]\) and then the polynomial automorphism \((\exp D)_{\star}\) of \(k^n\). In this note we present a general upper bound of mdeg \((\exp D)_{\star}\) in the case of a triangular derivation \(D\), and also show that this estimataion is exact.