This paper is devoted to the study of multigraded algebras and multigraded linear series. For an -graded algebra , we define and study its volume function , which computes the asymptotics of the Hilbert function of . We relate the volume function to the volume of the fibers of the global Newton–Okounkov body of . Unlike the classical case of standard multigraded algebras, the volume function is not a polynomial in general. However, in the case when the algebra has a decomposable grading, we show that the volume function is a polynomial with nonnegative coefficients. We then define mixed multiplicities in this case and provide a full characterization for their positivity. Furthermore, we apply our results on multigraded algebras to multigraded linear series. Our work recovers and unifies recent developments on mixed multiplicities. In particular, we recover results on the existence of mixed multiplicities for (not necessarily Noetherian) graded families of ideals and on the positivity of the multidegrees of multiprojective varieties.

中文翻译：

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多级代数和多级线性级数

本文致力于多级代数和多级线性级数的研究。为-分级代数，我们定义并研究它的体积函数，计算希尔伯特函数的渐近。我们将体积函数联系起来到整体牛顿-奥孔科夫体的纤维体积的。与标准多级代数的经典情况不同，体积函数一般而言不是多项式。然而，在代数的情况下具有可分解的分级，我们证明体积函数是具有非负系数的多项式。然后，我们在这种情况下定义混合多重性，并为其积极性提供完整的表征。此外，我们将多级代数的结果应用于多级线性级数。我们的工作恢复并统一了混合多样性的最新发展。特别是，我们恢复了关于（不一定是诺特式）理想分级族的混合多重性的存在以及多射影簇的多重度的正性的结果。