A Cautionary Tale

Abstract: Let $A \subseteq B$ be integral domains and $G$ be a totally
ordered Abelian group. D. Daigle has formulated certain hypotheses on
degree function $\deg : B \rightarrow G \cup \lbrace - \infty \rbrace$ so
that it is tame in characteristic zero, i.e., $\deg(D)$ is defined for all
$A$-derivations $D: B \rightarrow B$. This study is important because each
$D \in \der_k(B)$ for which $\deg(D)$ is defined, we can homogenize the
derivation which is an important and useful tool in the study of
$\G_a$-action on an algebraic variety.

In arbitrary characteristic, $\G_a$-action on an affine scheme $\spec(B)$
can be interpreted in terms of exponential maps on $B$. In this talk we
shall discuss analogous formulations of hypotheses on the degree function
so that $\deg(\phi)$ is defined for each $A$-linear exponential map $\phi$
on $B$. This talk is based on a joint work with N. Gupta.