We study a partial correspondence between two previously-studied analogues of Gröbner bases in the setting of algebras: namely subalgebra bases for quotients of polynomial rings and Khovanskii bases for valued algebras and domains. Our main motivation is to apply the concrete and computational aspects of subalgebra bases for quotient rings to the abstract theory of Khovanskii bases. Our perspective is that most interesting examples of Khovanskii bases can also be realized as subalgebra bases and vice-versa. As part of this correspondence, we extend the theory of subalgebra bases for quotients of polynomial rings to infinitely generated polynomial algebras and study conditions which make this theory effective. We also provide a computation of Newton-Okounkov bodies from the data of subalgebra bases for quotient rings, which illustrates how interpreting Khovanskii bases as subalgebra bases makes them amenable to existing computer algebra tools.

The main results of this paper establish a partial correspondence between two previously-studied analogues of Gröbner bases in the setting of algebras: namely, subalgebra (aka SAGBI) bases for quotients of polynomial rings and Khovanskii bases for valued algebras. We aim to bridge the gap between the concrete, computational aspects of the former and the more abstract theory of the latter. Our philosophy is that most interesting examples of Khovanskii bases can also be realized as subalgebra bases and vice-versa. We also discuss the computation of Newton-Okounkov bodies, illustrating how interpreting Khovanskii bases as subalgebra bases makes them more amenable to the existing tools of computer algebra.