From neural codes to homological invariants: regularity and projective dimension of polarized neural ideals
Trung Chau
TL;DR
This work determines the homological landscape of polarized neural ideals by providing sharp bounds and complete value ranges for the invariants $\mathrm{pd}$ and $\mathrm{reg}$ on $n$ neurons, with tighter bounds when generators have degree $n$. A central contribution is the recursive decomposition $I= x_n J+y_n K$ that ties a degree-$n$ polarized neural ideal to $(n-1)$-neuron ideals, enabling an inductive analysis. The authors then characterize when such ideals have linear resolution or linear quotients, showing that for degree-$n$ generation these properties are equivalent to analogous properties of $J$ and $K$ with a containment relation. Collectively, these results connect neural coding structures to classical combinatorial commutative algebra, yielding a practical recursive criterion for LR and LQ and illuminating the role of dominant monomial sets and Betti splittings in this context.
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 $n$.