Khovanskii Bases, Higher Rank Valuations, and Tropical Geometry

Given a finitely generated algebra $A$, it is a fundamental question whether $A$ has a full rank discrete (Krull) valuation $\mathfrak{v}$ with finitely generated value semigroup. We give a necessary and sufficient condition for this in terms of tropical geometry of $A$. In the course of this we introduce the notion of a Khovanskii basis for $(A,\mathfrak{v})$ which provides a framework for far extending Gröbner theory on polynomial algebras to general finitely generated algebras. In particular, this makes a direct connection between the theory of Newton--Okounkov bodies and tropical geometry, and toric degenerations arising in both contexts. We also construct an associated compactification of ${Spec}\,(A)$. Our approach includes many familiar examples such as the Gel'fand--Zetlin degenerations of coordinate rings of flag varieties as well as wonderful compactifications of reductive groups. We expect that many examples coming from cluster algebras naturally fit into our framework.