Affine surfaces with trivial Makar-Limanov invariant

Abstract

We study the class of 2-dimensional affine k-domains R satisfying ML ( R ) = k , where k is an arbitrary field of characteristic zero. In particular, we obtain the following result: Let R be a localization of a polynomial ring in finitely many variables over a field of characteristic zero. If ML ( R ) = K for some field K ⊂ R such that trdeg K R = 2 , then R is K-isomorphic to K [ X , Y , Z ] / ( X Y − P ( Z ) ) for some nonconstant P ( Z ) ∈ K [ Z ] .