Positivity of Mixed Multiplicities of Filtrations

The theory of mixed multiplicities of filtrations by $m$-primary ideals in a ring is introduced in a recent paper by Cutkosky, Sarkar and Srinivasan. In this paper, we consider the positivity of mixed multiplicities of filtrations. We show that the mixed multiplicities of filtrations must be nonnegative real numbers and give examples to show that they could be zero or even irrational. When $R$ is analytically irreducible, and $\mathcal{I}(1),\dots ,\mathcal{I}(r)$ are filtrations of $R$ by $m_R$-primary ideals, we show that all of the mixed multiplicities $e_R(\mathcal{I}(1{)}^{[d_1]},\dots ,\mathcal{I}(r{)}^{[d_r]};R)$ are positive if and only if the ordinary multiplicities $e_R(\mathcal{I}(i);R)$ for $1\le i\le r$ are positive. We extend this to modules and prove a simple characterization of when the mixed multiplicities are positive or zero on a finitely generated module.