الگوریتم منطقی سه گانه Rationally triangulable autmorphisms

A rational action of an algebraic group G, defined over the characteristic zero, algebraically-closed field k, on the affine space A”(k), is said to be trianguleble if coordinates x1, . . . , x,, can be chosen so that the induced automorphism on the coordinate ring has the form xi H cyixi + Fi(x, , . . . , xi_, ) with cyi in the nrultiplicative group of k. The action is said to be linear if there is a coordinate system on which it is effected by a linear change of variables, and tame if it lies in the group generated by the triangular and linear automorphisms. It is known that the automorphism group of A’(k) is the amalgamated free product of the groups of linear and triangular automorphisms, but it remains unknown whether these subgroups generate the automorphism group if n 2 3. Bass, in [ 11, and Popov, in [4], have given examples of actions of the additive group of k, denoted G,, on A”(k) which are neither linearizable nor triangulable. The structure theory of amalgamated products thus shows that the automorphism group cannot have this structure for n 2 3. Two approximations to tameness are the notions of stable tameness and rational triangulability. An action of G on A”(k) is staL$~ tame provided its extension to A”+‘” (k) by fixing the last m coordinates is tame, and ratiolzaC/y trianguiable if there are generators y, , . . . , y, of the field of rational functions so that each of the subfields k( y, , . . . , Yj) is invariant under the group of kautomorphisms of the rational function field induced by G. In [6], Smith showed that the examples of Popov are stably tame. It was asked in [l] whether every rational action of a unipotent group on affine space is rationally triangulable. This paper provides a necessary and sufficient condition for the rational triangulability of actions of the additive group of k on affine space. The criterion can be used to demonstrate the rational triangulability of all G, actions on A3(k), in particular those of [I] and [4], as well as to prove, for arbitrary n, that all G, actions are stably rationally triangulable (indeed they are rationally triangulable in the extension of the action to An + 1(k)).