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Polarization and depolarization of monomial ideals with application to multi-state system reliability. (English) Zbl 1441.13048

To study an ideal in a polynomial ring, it is a standard technique to deal instead with its initial ideal under suitable monomial ordering. This initial ideal is a monomial ideal and is relatively easier to handle. But if it is not squarefree, more than often, one will additionally apply the polarization tool to study the associated squarefree monomial ideal. Whence, one can borrow the strength of combinatorics, for instance, by studying the associated Stanley-Reisner complex or clutter. Another nice thing about polarization is that it can preserve many good algebraic properties of the original ideal. One drawback of applying the polarization, however, is that one has to count on a larger polynomial ring. And this increases the computational complexity in general.
The paper under review considers additionally the reverse process. Given a squarefree monomial ideal, the authors introduce the depolarization orders and use this tool to generate all depolarizations of this ideal. These newly generated ideals are called copolar in this paper.
Since copolar ideals have isomorphic lcm-lattices, and the projective dimension is bounded by the dimension of the ambient ring, this paper provides a new bound of the projective dimension of a general monomial ideal. It requires the application of both polarization and depolarization.
Furthermore, under some mild condition, a squarefree monomial ideal is copolar to some zero-dimensional monomial ideal. Since the latter is quasi-stable, one can then obtain the Castelnuovo-Mumford regularity and the projective dimension with ease.
The final section of this paper is devoted to the applications in the system reliability theory. Notice that the depolarization allows one to consider within a smaller polynomial ring. When coming to the coherent systems, it reduces the dimension and number of variables. Therefore, by the four examples exhibited in this section, it seems that the algebraic method here is quite practical.

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
05E40 Combinatorial aspects of commutative algebra
90B25 Reliability, availability, maintenance, inspection in operations research

Software:

Macaulay2
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References:

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