Neural codes via homological invariants of polarized neural ideals

TL;DR

This paper explores the homological invariants of polarized neural ideals derived from neural codes, linking algebraic properties to the topology of associated simplicial complexes and the geometry of the Hamming cube.

Contribution

It establishes bounds and characterizes extremal cases for the projective dimension and regularity of polarized neural ideals using topological and geometric insights.

Findings

Maximum regularity is 2n-1, achieved by codes missing an antipodal pair.

Projective dimension is at most 2n-3, with explicit codes attaining this bound.

Coordinate subcubes and complements are characterized by minimal invariants.

Abstract

For a neural code $\mathcal{C}\subseteq\mathbb{F}_2^n$, polarizing the canonical form generators of the neural ideal $J_{\mathcal{C}}$ yields a squarefree monomial ideal $\mathcal{P}(J_{\mathcal{C}})\subset k[x_1,\dots,x_n,y_1,\dots,y_n]$, the polarized neural ideal, and an associated simplicial complex $\Delta_{\mathcal{C}}$, the polar complex. We study the graded invariants $\operatorname{pd}(\mathcal{P}(J_{\mathcal{C}}))$ and $\operatorname{reg}(\mathcal{P}(J_{\mathcal{C}}))$ via the topology of $\Delta_{\mathcal{C}}$, showing that simple geometric features of the Hamming cube $\mathbb{F}_2^n$ (with Hamming distance) organize their extremal behavior. We prove $\operatorname{reg}(\mathcal{P}(J_{\mathcal{C}}))\le 2n-1$, with equality precisely when $\mathcal{C}$ is obtained from $\mathbb{F}_2^n$ by deleting an antipodal pair. Using connectedness properties of induced subcomplexes of $\Delta_{\mathcal{C}}$, we obtain $\operatorname{pd}(\mathcal{P}(J_{\mathcal{C}}))\le 2n-3$, and we give an explicit family of codes attaining equality, each consisting of antipodal pairs. At the opposite end, we identify the cube geometry behind the smallest values: $\operatorname{reg}(\mathcal{P}(J_{\mathcal{C}}))=1$ forces $\mathcal{C}$ to be a coordinate subcube of $\mathbb{F}_2^n$, while $\operatorname{pd}(\mathcal{P}(J_{\mathcal{C}}))=0$ forces $\mathcal{C}$ to be the complement of one. Finally, we construct families realizing large regions of the $(\operatorname{pd},\operatorname{reg})$-plot for fixed $n$.