arXiv subjects

N. M. Artemov

Publications and source records attributed to N. M. Artemov.

3 recordsLinked to original sources

arXiv2026

Ramification of the volume function close to asymptotic hyperplanes. Monodromy of boundary singularities

It is shown that the discriminants of boundary singularities introduced by V.~I.~Arnold naturally arise in integral geometry. They represent the irregularity sets of the volume function. This function maps an affine hyperplane to the volume of the part of a domain in the Euclidean space cut off by this hyperplane. For isolated boundary singularities, the role of the boundary is shown to be related to the hyperplane at infinity in the projective space. Versions of the Milnor fiber bundle adapted to the present problem are defined and studied. Analogs of the local even Petrovsky class are introduced and computed. It is proved that the volume function is locally finite-sheeted only if these classes are trivial. Such components of the complements to the discriminants (quasilacunas) are listed for simple boundary singularities --- those having no moduli in the classification. The obtained results establish new obstructions to the algebraic integrability of domains in a Euclidean space.

math.AG↗

arXiv2025

Matrix representation of Picard--Lefschetz--Pham theory near the real plane in $\mathbb{C}^2$

A matrix formalism is proposed for computations based on Picard--Lefschetz theory in a 2D case. The formalism is essentially equivalent to the computation of the intersection indices necessary for the Picard--Lefschetz formula and enables one to prove non-trivial topological identities for integrals depending on parameters. We introduce the universal Riemann domain $\tilde U$, i.e. a sort of ``compactification'' of the universal covering space $\tilde U_2$ over a small tubular neighborhood $N\mathbb{R}^2$ of $\mathbb{R}^2\backslashσ$ in $\mathbb{B}\subset\mathbb{C}^2$, where $\mathbb{B}\subset\mathbb{C}^2$ is a big ball, and $σ$ is a one-dimensional complex analytic set (the set of singularities). We compute the Picard-Lefschetz monodromy of the relative homology group of the space $\tilde U$ modulo the singularities and the boundary for the standard local degenerations of type $P_1 ,P_2,P_3$ in Pham's [1] notations and for more complicated configurations in $\mathbb{C}^2$. We consider this homology group as a module over the group ring of the $π_1((N\mathbb{R}^2 \cap \mathbb{B})\backslashσ)$ over $\mathbb{Z}$. The results of the computations are presented in the form of a matrix of the monodromy operator calculated in a certain natural basis. We prove an ``inflation'' theorem, which states that the integration surfaces of interest (i.e.\ the elements of the homology group $H_2(\tilde U_2,\tilde{\partial \mathbb{B}})$) (the surfaces in the branched space possibly passing through singularities) are injectively mapped to the group $H_2(\tilde U,\tilde U'\cup\tilde{\partial \mathbb{B}})$ (the surfaces avoiding the singularities). The matrix formalism obtained describes the behaviour of integrals depending on parameters and can be applied to the study of Wiener-Hopf method in two complex variables.

math-ph↗

arXiv2022

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

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.

math.AG↗