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Betti numbers of inductively pierced codes
Hugh Geller, · 2026-05-26 · via math updates on arXiv.org

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Abstract:Neural ideals were introduced by Curto, Itskov, et al as an algebraic tool to study neural codes. In this paper, we use the notion of polarization introduced by Güntürkün, Jeffries, and Sun to compute Betti numbers of inductively pierced neural codes. We prove that quadratically generated neural ideals are inductively pierced if and only if they have regularity 2. Further, we demonstrate how the Betti numbers yield information on the number and type of piercings of the original code.
This work shows the utility of algebraic invariants of the neural ideal in detecting geometric features of the associated receptive fields.
Comments: 34 pages, 5 figures
Subjects: Commutative Algebra (math.AC)
MSC classes: Primary: 13P25, Secondary: 92-10, 92C99, 13D02, 13F55
Cite as: arXiv:2605.24181 [math.AC]
  (or arXiv:2605.24181v1 [math.AC] for this version)
  https://doi.org/10.48550/arXiv.2605.24181

arXiv-issued DOI via DataCite (pending registration)

Submission history

From: Rebecca R.G. [view email]
[v1] Fri, 22 May 2026 20:09:06 UTC (1,430 KB)