From neural codes to homological invariants: regularity and projective dimension of polarized neural ideals
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908671588433920 |
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| author | Chau, Trung |
| author_facet | Chau, Trung |
| contents | Neural codes form an algebraic framework to study the nervous system, and understanding neural codes is a key goal of mathematical neuroscience. Neural rings and ideals are the tools connecting neuroscience and commutative algebra. In this article, we study the projective dimension and (Castelnuovo-Mumford) regularity of polarized neural ideals on $n$ neurons. Particularly, we find all the possible values for these two invariants. Moreover, we characterize when these ideals have linear resolution or linear quotients, assuming that they are generated in degree $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19043 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From neural codes to homological invariants: regularity and projective dimension of polarized neural ideals Chau, Trung Commutative Algebra 13F20, 13P25, 13L99, 92B20 Neural codes form an algebraic framework to study the nervous system, and understanding neural codes is a key goal of mathematical neuroscience. Neural rings and ideals are the tools connecting neuroscience and commutative algebra. In this article, we study the projective dimension and (Castelnuovo-Mumford) regularity of polarized neural ideals on $n$ neurons. Particularly, we find all the possible values for these two invariants. Moreover, we characterize when these ideals have linear resolution or linear quotients, assuming that they are generated in degree $n$. |
| title | From neural codes to homological invariants: regularity and projective dimension of polarized neural ideals |
| topic | Commutative Algebra 13F20, 13P25, 13L99, 92B20 |
| url | https://arxiv.org/abs/2511.19043 |