arXiv subjects

Vladimir Shemyakov

Publications and source records attributed to Vladimir Shemyakov.

4 recordsLinked to original sources

arXiv2026

Second-Order Differential Equations and Sums of Squares of Cauchy Kernels with Finitely Many Zeros

We study finite-order meromorphic functions representable as absolutely convergent sums of squares of Cauchy kernels and having only finitely many zeros. By earlier work of Baranov and the author, such functions admit a representation $f=P/g^2$, where $P$ is a polynomial and $g$ is entire, satisfying the differential equation $ Pg''-P'g'+Qg=0, $ where $Q$ is a polynomial. We show that the zeros of $g$ asymptotically accumulate along the Stokes rays. If $\mathrm{deg}\ Q>\mathrm{deg}\ P$, they approach these rays in the Euclidean metric, whereas in the borderline case $\mathrm{deg}\ Q=\mathrm{deg}\ P$ one obtains in general only localization in logarithmic neighborhoods of the Stokes rays, and this is sharp. We then characterize the existence of a decomposition $ P/g^2=\sum c_n (z-t_n)^{-2} $ in terms of the sectorial behavior of $g$ and, equivalently, in terms of the Laine condition for the corresponding Schwarzian equation. Finally, for fixed $P$ and fixed order, we identify the resulting families, modulo the natural equivalence relation, with finite-dimensional affine algebraic varieties.

math.CV↗

arXiv2024

Zeros of meromorphic functions of the form $\sum\limits_n \dfrac{c_n}{(z-t_n)^2}$

We study zeros distribution for meromorphic functions of the form $\sum\limits_n \dfrac{c_n}{(z-t_n)^2}$, where $\sum\limits_n \dfrac{|c_n|}{|t_n|^2} <\infty$. We prove an analog of the classical Keldysh theorem and discuss a relation between zero-free functions of this form and second order differential equtions with polynomial coefficients.

math.CV↗

arXiv2024

The zeta function of zeros and poles of a meromorphic function, the criterion for the absence of zeros, and its application to sums of Cauchy kernels

A connection between the zeta functions of zeros and poles of a meromorphic function has been established, and using it, a criterion for the absence of zeros has been derived. Sufficient conditions for the existence of zeros of sums of Cauchy kernels have been obtained, including those based on geometric conditions on the poles.

math.CV↗

arXiv2024

The infinitesimal behavior of the sum of Cauchy kernels and its derivative at infinity

In analysis, it's often useful to know the value of a function at infinity, this operation possesses pleasant properties. However, even when the limit does not exist, some intuitive considerations may suggest that the function still assumes a specific value at infinity in a certain sense. In Nevanlinna theory, all objects are studied on average, i.e., their integrals, hence the integral interpretation of the concept of convergence to a limit is beneficial for the theory of meromorphic functions. This is precisely the focus of this work, applied to sums of Cauchy kernels and their derivatives.

math.CV↗