The LFED and LNED Conjectures for Laurent Polynomial Algebras

Let $R$ be an integral domain of characteristic zero, $x=(x_1, x_2, ..., x_n)$ $n$ commutative free variables, and ${\mathcal A}_n:=R[x^{-1}, x]$, i.e., the Laurent polynomial algebra in $x$ over $R$. In this paper we first classify all locally finite or locally nilpotent $R$-derivations and $R$-$\mathcal E$-derivations of ${\mathcal A}_n$, where by an $R$-$\mathcal E$-derivation of ${\mathcal A}_n$ we mean an $R$-linear map of the form $\operatorname{Id}_{{\mathcal A}_n}-\phi$ for some $R$-algebra endomorphism $\phi$ of ${\mathcal A}_n$. In particular, we show that ${\mathcal A}_n$ has no nonzero locally nilpotent $R$-derivations or $R$-$\mathcal E$-derivations. Consequently, the LNED conjecture proposed in [Z4] for ${\mathcal A}_n$ follows. We then show some cases of the LFED conjecture proposed in [Z4] for ${\mathcal A}_n$. In particular, we show that both the LFED and LNED conjectures hold for the Laurent polynomial algebras in one or two commutative free variables over a field of characteristic zero.