Polarization and depolarization of monomial ideals with application to multi-state system reliability

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.