On global non-oscillation of linear ordinary differential equations with polynomial coefficients (Q305404)

Consider a linear differential equation NEWLINE\[NEWLINE P_k(z)y^{(k)} +P_{k-1}(z)y^{(k-1)}+\dots +P_0(z)y=0 ,\eqno(1)NEWLINE\]NEWLINE where \(P_k\), \(P_{k-1}, \dots, P_0\) are polynomials with \(\mathrm{GCD}(P_k, P_{k-1}, \dots, P_0)=1\). Let \(S\) be the set of all singular points of (1) in \(\mathbb{CP}^1\), i.e. the set of all roots of \(P_k\). We write \(d=\mathrm{card }S\). A number \(\lambda\in \mathbb{C}\) is called a characteristic exponent at a point \(p\in \mathbb{C}\), if, in a punctured disc \(0<|z-p|<\varepsilon\), \(\varepsilon>0\), there exists a local solution of (1) of the form \(y(z)=(z-p)^\lambda\sum_{m=0}^\infty a_m(z-p)^m\), \(a_0\neq 0\).NEWLINENEWLINEA system \(\bar C:\{ C_j\}_{j=1}^{d-1}\) of smooth Jordan curves in \(\mathbb{CP}^1\), each of them connecting a pair of distinct singular points, is called an admissible cut for equation (1) if and only if: a) for any \(i\neq j\), the intersection \(C_i\cap C_j\) is either empty or consists of their common endpoints; b) the complement \(\mathbb{CP}^1\setminus \bigcup_j C_j\) is contractible; c) each \(C_j\) has a well-defined tangent vector at each of its two endpoints.NEWLINENEWLINEThe main result of the paper is the following criterion.NEWLINENEWLINETheorem 8. Suppose that equation (1) has only Fuchsian singularities. Each nontrivial solution has finitely many zeros in \(\mathbb{CP}^1\setminus \bar C\) if and only if at each singular point all distinct characteristic exponents have pairwise distinct real parts.NEWLINENEWLINEThe proof uses ideas of the Wiman-Valiron theory.