The LFED and LNED Conjectures for Laurent Polynomial Algebras

Abstract

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 ${\mathrm{Id}}_{{\mathcal{A}}_n}-\varphi$ for some $R$-algebra endomorphism $\varphi$ 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.

17 pages

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• Images of Ideals under Derivations and $\mathcal{E}$-Derivations of Univariate Polynomial Algebras over a Field of Characteristic Zero

Cited by in corpus (1)

• Images of Ideals under Derivations and $\mathcal{E}$-Derivations of Univariate Polynomial Algebras over a Field of Characteristic Zero