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.

中文翻译：

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多度数什么时候是正数？

设 $k$ 是一个任意域，$P = P_k^{m_1} \times_k \cdots \times_k P_k^{m_p}$ 是 $k$ 上的多投影空间，$X \subseteq P$ 是$P$。我们为$X$ 的多度数的正性提供充要条件。作为我们方法的结果，我们表明当 $X$ 不可约时，多度的支持形成离散代数多拟阵。在代数术语中，我们刻画了标准多级代数在 Artinian 局部环上的混合多重性的正性，并将其应用于理想的混合多重性的积极性。此外，我们使用我们的结果在组合代数几何的背景下恢复文献中的几个结果。