Monomial polarization and depolarization of abstract simplicial complexes
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911083710644224 |
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| author | López-Antón, Víctor M. Munarriz-Senosiain, Pablo Pascual-Ortigosa, Patricia Sáenz-de-Cabezón, Eduardo |
| author_facet | López-Antón, Víctor M. Munarriz-Senosiain, Pablo Pascual-Ortigosa, Patricia Sáenz-de-Cabezón, Eduardo |
| contents | We translate the operations of polarization and depolarization from monomial ideals in a polynomial ring to abstract simplicial complexes. As a result, we explicitly describe the relation between the Koszul simplicial complex of a monomial ideal and that of its polarization. Using the simplicial translation of depolarization we propose a way to reduce a simplicial complex to a smaller one with the same homology. This type of reduction, that can be interpreted as non-elementary collapse, can be used as a pre-process step for algorithms on simplicial complexes. We apply this methodology to the efficient computation of the Alexander dual of abstract simplicial complexes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_22497 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monomial polarization and depolarization of abstract simplicial complexes López-Antón, Víctor M. Munarriz-Senosiain, Pablo Pascual-Ortigosa, Patricia Sáenz-de-Cabezón, Eduardo Commutative Algebra Algebraic Topology 55U10, 13F55 We translate the operations of polarization and depolarization from monomial ideals in a polynomial ring to abstract simplicial complexes. As a result, we explicitly describe the relation between the Koszul simplicial complex of a monomial ideal and that of its polarization. Using the simplicial translation of depolarization we propose a way to reduce a simplicial complex to a smaller one with the same homology. This type of reduction, that can be interpreted as non-elementary collapse, can be used as a pre-process step for algorithms on simplicial complexes. We apply this methodology to the efficient computation of the Alexander dual of abstract simplicial complexes. |
| title | Monomial polarization and depolarization of abstract simplicial complexes |
| topic | Commutative Algebra Algebraic Topology 55U10, 13F55 |
| url | https://arxiv.org/abs/2507.22497 |