Let $F$ be an algebraically closed field and suppose that the torus $(F^*)^n$ acts on the Laurent polynomial ring $L$ in $n$ variables $X_1, \dots, X_n$ defined by $X_i \dashrightarrow a_iX_i$ for suitable scalars $a_i$.
Assume that the $a_i$ generate a subgroup of rank $n$ in $F^*$.
This action partitions the set of maximal ideals of $L$ into orbits.
For $f, g$ in $L$ say that an orbit $O$ 'occurs' if $f$ and $g$ lie in suitable maximal ideals $P$ and $Q$ respectively belonging to $O$.
In this situation, for given $f$ and $g$ that are not monomials is it true that only finitely many orbits 'occur' ?
