Neural Ideal Preserving Homomorphisms

The neural ideal of a binary code \(\mathbb{C} \subseteq \mathbb{F}_2^n\) is an ideal in \(\mathbb{F}_2[x_1,\ldots, x_n]\) closely related to the vanishing ideal of \(\mathbb{C}\). The neural ideal, first introduced by Curto et al, provides an algebraic way to extract geometric properties of realizations of binary codes. In this paper we investigate homomorphisms between polynomial rings \(\mathbb{F}_2[x_1,\ldots, x_n]\) which preserve all neural ideals. We show that all such homomorphisms can be decomposed into a composition of three basic types of maps. Using this decomposition, we can interpret how these homomorphisms act on the underlying binary codes. We can also determine their effect on geometric realizations of these codes using sets in \(\mathbb{R}^d\). We also describe how these homomorphisms affect a canonical generating set for neural ideals, yielding an efficient method for computing these generators in some cases.