Newton-Okounkov theory in an abstract setting (Q2028007)

The purpose of this paper is to construct Newton-Okounkov bodies for graded cancellative torsion-free semigroups \(S\), thanks to the fact that they can be embedded in \(\mathbb{R}^n\), by extending the construction of \textit{K. Kaveh} and \textit{A. G. Khovanskii} [Ann. Math. (2) 176, No. 2, 925--978 (2012; Zbl 1270.14022)]. This is indeed possible, and many of the most important properties of this remain true in the more algebraic context, as Theorem 3.9 ensures; remarkably, the volume of the Newton-Okounkov body associated to \(S\) governs the growth rate of the Hilbert function of \(S\). Moreover, the authors manage to generalize in Theorem 5.7 the equality between volumes of Newton-Okounkov bodies and volumes of line bundles for the bodies associated to \(S\)-graded algebras. The paper finishes with an appendix explaining the filtered Newton-Okounkov bodies in the setting of the paper under consideration.