2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/217413Let $S$ be a finitely generated standard multigraded algebra over an Artinian local ring $A$; $M$ a finitely generated multigraded $S$-module. This paper answers to the question when mixed multiplicities of $M$ are positive and characterizes them in terms of lengths of $A$-modules. As an application, we get results on mixed multiplicities of ideals (Proposition 4.4 and Theorem 4.5), and recover the results of Risler and Teissier [Te](see Remark 4.8), Trung and Verma [TV](see Remark 4.6).23 pagesCommutative Algebra13H15text