  Let $k$ be an arbitrary field, $P = P_k^{m_1} \times_k \cdots \times_k
P_k^{m_p}$ be a multiprojective space over $k$, and $X \subseteq P$ be a closed
subscheme of $P$. We provide necessary and sufficient conditions for the
positivity of the multidegrees of $X$. As a consequence of our methods, we show
that when $X$ is irreducible, the support of multidegrees forms a discrete
algebraic polymatroid. In algebraic terms, we characterize the positivity of
the mixed multiplicities of a standard multigraded algebra over an Artinian
local ring, and we apply this to the positivity of mixed multiplicities of
ideals. Furthermore, we use our results to recover several results in the
literature in the context of combinatorial algebraic geometry.
