Geometry, combinatorics, and algebra of inductively pierced codes

Convex neural codes are combinatorial structures describing the intersection pattern of a collection of convex sets. Inductively pierced codes are a particularly nice subclass of neural codes introduced in the information visualization literature by Stapleton et al. in 2011 and to the convex codes literature by Gross et al. in 2016. Here, we show that all inductively pierced codes are nondegenerate convex codes and nondegenerate hyperplane codes. In particular, we prove that a $k$-inductively pierced code on $n$ neurons has a convex realization with balls in $\mathbb R^{k+1}$ and with half spaces in $\mathbb R^{n}$. We characterize the simplicial and polar complexes of inductively pierced codes, showing the simplicial complexes are disjoint unions of vertex decomposable clique complexes and that the polar complexes are shellable. In an earlier version of this preprint, we gave a flawed proof that toric ideals of $k$-inductively pierced codes have quadratic Gr\"obner bases under the term order induced by a shelling order of the polar complex of $\mathcal C$. We now state this as a conjecture, inspired by computational evidence.