Neural Ideal Preserving Homomorphisms

TL;DR

This paper studies homomorphisms between polynomial rings that preserve neural ideals of binary codes, revealing their structure and impact on codes and geometric realizations, with implications for efficient computation.

Contribution

It characterizes neural ideal-preserving homomorphisms as compositions of basic maps and analyzes their effects on codes and realizations, providing new insights and computational methods.

Findings

All neural ideal-preserving homomorphisms decompose into three basic types.

These homomorphisms influence the underlying binary codes and their geometric realizations.

The paper offers an efficient method to compute generators of neural ideals under these homomorphisms.

Abstract

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.