Our construction of an integrable system uses Anderson's toric degeneration as a key ingredient, so in this section we both review and slightly expand on his exposition in [An10]. First we recall how the semigroup algebra C[S] of R is related to the **data** of the extended valuationṽ. The link comes from a filtration of R defined usingṽ as follows. Let
R ≥(m,u) := {f ∈ R |ṽ(f ) ≥ (m, u) or f = 0}
and similarly for R >(m,u) . By the definition of the valuation v and the definition of the order (7.3) on N × Z n , each R ≥(m,u) (and R >(m,u) ) is a (finite-dimensional) vector subspace of R. Moreover, again by properties of valuations, for any (m, u) and (m ′ , u ′ ), we have
R ≥(m,u) · R ≥(m ′ ,u ′ ) ⊆ R ≥(m+m ′ ,u+u ′ ) .
We can now define the associated graded ring grR with respect to this filtration as
grR := (m,u) R ≥(m,u) /R >(m,u) .
This is naturally an S-graded ring. In fact, sinceṽ has one-dimensional leaves, the spaces R ≥(m,u) /R >(m,u) have dimension 1 precisely when (m, u) ∈ S, and is 0 otherwise. Since the homogeneous elements of grR are not zero divisors, one shows that grR is isomorphic to the semigroup algebra C[S] (see [BG09,Remark 4.13]).