Neural codes via homological invariants of polarized neural ideals

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 $n$.