On the mixed multiplicities of multi-graded fiber cones (Q1012441)

Let \((A,\mathfrak{m})\) be a noetherian local ring and let \(M\) be a finitely generated \(R\)-module. Let \(J\) be an \(\mathfrak{m}\)-primary ideal and \(I_1,\dots, I_s\) be ideals of \(A\). In this paper the authors answer when mixed multiplicities of the multi-graded fiber cone \[ \bigoplus_{n_1\dots n_s\geq 0}I_1^{n_1}\dots I_s^{n_s}M/JI_1^{n_1}\dots I_s^{n_s}M \] are positive and characterize them in terms of the length of modules. This generalizes the work of \textit{D. Rees} [J. Lond. Math. Soc., II. Ser. 29, 397--414 (1984; Zbl 0572.13005)], where all ideals are \(\mathfrak{m}\)-primary.