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

Abstract

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 $P{g}^{\prime \prime }-P^{\prime }{g}^{\prime }+Qg=0,$ 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.