Three invariants of geometrically vertex decomposable ideals
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912173113999360 |
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| author | Nguyen, Thai Thanh Rajchgot, Jenna Van Tuyl, Adam |
| author_facet | Nguyen, Thai Thanh Rajchgot, Jenna Van Tuyl, Adam |
| contents | We study three invariants of geometrically vertex decomposable ideals: the Castelnuovo-Mumford regularity, the multiplicity, and the $a$-invariant. We show that these invariants can be computed recursively using the ideals that appear in the geometric vertex decomposition process. As an application, we prove that the $a$-invariant of a geometrically vertex decomposable ideal is non-positive. We also recover some previously known results in the literature including a formula for the regularity of the Stanley--Reisner ideal of a pure vertex decomposable simplicial complex, and proofs that some well-known families of ideals are Hilbertian. Finally, we apply our recursions to the study of toric ideals of bipartite graphs. Included among our results on this topic is a new proof for a known bound on the $a$-invariant of a toric ideal of a bipartite graph. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_08541 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Three invariants of geometrically vertex decomposable ideals Nguyen, Thai Thanh Rajchgot, Jenna Van Tuyl, Adam Commutative Algebra Combinatorics 13P10, 14M25, 05E40 We study three invariants of geometrically vertex decomposable ideals: the Castelnuovo-Mumford regularity, the multiplicity, and the $a$-invariant. We show that these invariants can be computed recursively using the ideals that appear in the geometric vertex decomposition process. As an application, we prove that the $a$-invariant of a geometrically vertex decomposable ideal is non-positive. We also recover some previously known results in the literature including a formula for the regularity of the Stanley--Reisner ideal of a pure vertex decomposable simplicial complex, and proofs that some well-known families of ideals are Hilbertian. Finally, we apply our recursions to the study of toric ideals of bipartite graphs. Included among our results on this topic is a new proof for a known bound on the $a$-invariant of a toric ideal of a bipartite graph. |
| title | Three invariants of geometrically vertex decomposable ideals |
| topic | Commutative Algebra Combinatorics 13P10, 14M25, 05E40 |
| url | https://arxiv.org/abs/2311.08541 |