I will start with a motivation by briefly talking about Grobner basis for ideals
and SAGBI basis for subalgebras. Then continue on admissible orders on Zn (due to
  Robbiano). The orders on Zn will induce ordering on the monomials of Laurent
polynomial algebra, k[x1] = k[x1 1 , . . . , x1 n ], with base field k. We
  restrict ourselves to initial algebras of ring of multiplicative invariants,
   k[x1]G for G  GLn(Z), and determine cardinality of the distinct initial
                              algebras of k[x1]G.