Lacunas and ramification of volume functions at simple asymptotic hyperplanes and monodromy of boundary function singularities

TL;DR

This paper investigates the behavior of volume functions at special hyperplanes, linking their regularity and singularities to boundary function singularities and monodromy, extending classical results to asymptotic cases.

Contribution

It introduces a new analysis of volume functions at asymptotic hyperplanes, defining Petrovsky classes and lacunas, and relates these to boundary function singularities and monodromy groups.

Findings

Defined and computed local Petrovsky classes for asymptotic hyperplanes.

Identified conditions under which volume functions form lacunas.

Calculated local monodromy groups for volume functions.

Abstract

The volume function defined by a domain in Euclidean space $\mathbb{R}^n$ is the function on the space of affine hyperplanes equal to volumes cut by these hyperplanes from the domain. The study of these functions originates from the works of Archimedes and Newton and is closely related to the theory of lacunas of hyperbolic partial differential equations.The volume functions are regular at the hyperplanes of general position with respect to the boundary of the cut domain. We study their behavior at the non-regular hyperplanes, which are either tangent to the boundary of the domain at its finite points or have asymptotic direction. In both cases the local regularity of the restriction of the volume function to a local connected component of the set of regular planes depends on the triviality of a certain relative homology class, the (generalized) even Petrovsky class. In the first case (of finite tangencies) the study of these classes and enumeration of components of regularity (so called local lacunas) at simple singularities of wave fronts was essentially done by V. A. Vassiliev; it is formulated in terms of deformations of simple real function singularities. We show that the analogous study for asymptotic hyperplanes is related in the same way with the study of boundary function singularities. We define and calculate local Petrovsky classes for this case and find lacunas, i.e. the local components of the complement of corresponding discriminant in which all local Petrovsky classes are trivial. If component is not lacuna we prove that the corresponding volume function cannot be algebraic. Also we calculate the local monodromy groups, describing the ramification of volume functions.