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- W2900296440 abstract "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. Finally, we prove that toric ideals of $k$-inductively pierced codes have quadratic Grobner bases, generalizing and strengthening a previous result that toric ideals of 1-inductively pierced codes have quadratic generating sets. The order on codewords we use to generate a term order for this Grobner basis coincides with a shelling order of the polar complex of $mathcal C$." @default.
- W2900296440 created "2018-11-16" @default.
- W2900296440 creator A5078411223 @default.
- W2900296440 date "2018-11-12" @default.
- W2900296440 modified "2023-09-27" @default.
- W2900296440 title "Inductively pierced codes and neural toric ideals" @default.
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