**Parameter-dependent linear ordinary differential\
equations and topology of domains**

Vyacheslav M. Boyko$^{\dag 1}$, Michael Kunzinger$^{\ddag 2}$ and Roman O. Popovych$^{\dag\ddag\S 3}$

[*${}^\dag$Institute of Mathematics of National Academy of Sciences of Ukraine,\
$\phantom{{}^\dag}$3 Tereshchenkivska Str., Kyiv-4, 01601 Ukraine\
$^\ddag$Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria\
$^\S$Mathematical Institute, Silesian University in Opava, Na Rybníčku 1, 746 01 Opava,\
$\phantom{^\S}$Czech Republic*]{}\
E-mail: $^1$boyko@imath.kiev.ua, $^2$michael.kunzinger@univie.ac.at, $^3$rop@imath.kiev.ua

Introduction {#sec:intro}
============

The solution theory of $p$th order linear ordinary differential equations (ODEs) $$\label{eq:ode_no_parameter}
\sum_{i=0}^p g^i(x)\frac{{\rm d}^iu}{{\rm d} x^i}=f(x),$$ where $u$ is the unknown function, the independent variable $x$ varies in some interval $(a,b)$, $g^0,\dots,g^p,f\in {\rm C}\big((a,b)\big)$ and $g^p(x)\not=0$ for all $x\in (a,b)$, is a classical subject that is a part of most textbooks in the field (e.g., [@Amann1990; @Hartman2002; @Walter1998]). The set of all (classical) solutions to the homogeneous equation ($f=0$) forms a $p$-dimensional vector subspace $V$ of ${\rm C}^p\big((a,b)\big)$, while the solution space of is an affine subspace obtained by translating $V$ by any particular solution of . Any tuple of $p$ linearly independent solutions $(\varphi^1,\dots,\varphi^p)$ of the homogeneous equation is called a fundamental set of solutions, and setting $\varphi_{s-1}:={\rm d}^{s-1}\varphi/{\rm d} x^{s-1}$, $s=1,\dots,p$, we write $$\label{eq:def_wronskian}
\mathrm W(\varphi^1,\dots,\varphi^p) := \det
\begin{pmatrix}
