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We initiate a combinatorial study of Newton-Okounkov functions on toricvarieties with an eye on the rationality of asymptotic invariants of linebundles. In the course of our efforts we identify a combinatorial conditionwhich ensures a controlled behavior of the appropriate Newton-Okounkov functionon a toric surface. Our approach yields the rationality of many Seshadriconstants that have not been settled before.

Toric Newton-Okounkov functions with an application to the rationality of certain Seshadri constants on surfaces