A. Duncan, Z. Reichstein, SAGBI bases in rings of invariant Laurent polynomials, to appear in Proceedings of the AMS, Posted February 2008.

Abstract: Let $k$ be a field, L_n = k[x_1^{\pm 1}, \dots, x_n^{\pm 1}] be the Laurent polynomial ring in n variables and G be a group of k-algebra automorphisms of L_n. We give a necessary and sufficient condition for the ring of invariants L_n^G to have a SAGBI basis. We show that if this condition is satisfied then L_n^G has a SAGBI basis relative to any choice of coordinates in L_n and any term order.