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arXiv · 2511.19043

From neural codes to homological invariants: regularity and projective dimension of polarized neural ideals

Trung Chau
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Abstract

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 $n$.

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Trung Chau. 2025-11-24. From neural codes to homological invariants: regularity and projective dimension of polarized neural ideals. https://arxiv.org/abs/2511.19043

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