  We initiate a combinatorial study of Newton-Okounkov functions on toric
varieties with an eye on the rationality of asymptotic invariants of line
bundles. In the course of our efforts we identify a combinatorial condition
which ensures a controlled behavior of the appropriate Newton-Okounkov function
on a toric surface. Our approach yields the rationality of many Seshadri
constants that have not been settled before.

Algebraic Geometry;Commutative Algebra;Combinatorics