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arXiv · 2605.24181

Betti numbers of inductively pierced codes

Hugh Geller·Rebecca R. G.·Alexandra Seceleanu·Nora Youngs
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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 G\"{u}nt\"{u}rk\"{u}n, 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.

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Hugh Geller, Rebecca R. G., Alexandra Seceleanu, Nora Youngs. 2026-05-22. Betti numbers of inductively pierced codes. https://arxiv.org/abs/2605.24181

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