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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. [ABSTRACT FROM AUTHOR]

This paper investigates the existence and properties of a Bernstein–Sato functional equation in nonregular settings. In particular, we construct D-modules in which such formal equations can be studied. The existence of the Bernstein–Sato polynomial for a direct summand of a polynomial over a field is proved in this context. It is observed that this polynomial can have zero as a root, or even positive roots. Moreover, a theory of V-filtrations is introduced for nonregular rings, and the existence of these objects is established for what we call differentially extensible summands. This family of rings includes toric, determinantal, and other invariant rings. This new theory is applied to the study of multiplier ideals and Hodge ideals of singular varieties. Finally, we extend known relations among the objects of interest in the smooth case to the setting of singular direct summands of polynomial rings. [ABSTRACT FROM AUTHOR]