Polarization is a powerful technique in algebra which provides combinatorial tools to study algebraic invariants of monomial ideals. We study the reverse of this process, depolarization which leads to a family of ideals which share many common features with the original ideal. Given a squarefree monomial ideal, we describe a combinatorial method to obtain all its depolarizations, and we highlight their similar properties such as the graded Betti numbers. We show that even though they have many similar properties, their differences in dimension make them distinguishable in applications in system reliability theory. In particular, we apply polarization and depolarization tools to study the reliability of multistate coherent systems via binary systems and vice versa. We use depolarization as a tool to reduce the dimension and the number of variables in coherent systems.

中文翻译：

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单项式理想的极化和去极化及其在多态系统可靠性中的应用

极化是代数中的一项强大技术，它提供了组合工具来研究单项理想的代数不变量。我们研究了这一过程的反向过程，即去极化，这导致了一系列理想，这些理想与原始理想具有许多共同特征。给定无平方单项式理想，我们描述了一种组合方法来获得其所有的去极化，并强调了它们的相似属性，例如渐变的贝蒂数。我们证明，即使它们具有许多相似的特性，但它们在尺寸上的差异也使它们在系统可靠性理论的应用中具有明显的区别。特别是，我们使用极化和去极化工具通过二进制系统研究多态相干系统的可靠性，反之亦然。