论文标题
多边形何时呈阳性？
When are multidegrees positive?
论文作者
论文摘要
令$ k $为任意字段，$ p = p_k^{m_1} \ times_k \ cdots \ times_k p_k^{m_p} $是$ k $上的多重点空间，$ x \ x \ subseteq p $ be a nocle of Subscheme of a subscheme $ p $。我们为$ x $的多边形的积极性提供了必要的条件。由于我们的方法，我们表明，当$ x $不可记述时，多视频的支持形成了离散的代数多膜体。用代数术语，我们表征了标准多式代数在Artinian局部环上的混合多重性的积极性，并且我们将其应用于理想混合多重性的阳性。此外，我们利用结果在组合代数几何形状的背景下恢复文献中的几个结果。
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.
