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I will start with a motivation by briefly talking about Gr¨obner 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[x±1] = k[x±1
1 , . . . , x±1
n ], with base field k. We
restrict ourselves to initial algebras of ring of multiplicative invariants, k[x±1]G for
G  GLn(Z), and determine cardinality of the distinct initial algebras of k[x±1]G.
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