Polarization of neural rings

Abstract

Neural rings and ideals, as introduced by Curto, Itskov, Veliz-Cuba, and Youngs, are useful algebraic tools for organizing the combinatorial information of neural codes. However, they are neither graded nor local, and consequently, many algebraic approaches to understanding such rings fail. In this paper, we introduce an operation, called “polarization,” that allows us to relate neural ideals with squarefree monomial ideals, which are very well studied and known for their nice behavior in commutative algebra. We use this notion of polarization to construct free resolutions of neural ideals. We also relate the structure of polarizations of neural ideals to combinatorial properties of codes.