Graphs of zeros of analytic families

Abstract:

Let $\mathcal {F}:=\{f_{\lambda }\}$ be a family of holomorphic functions in a domain $D\subset \mathbb {C}$ depending holomorphically on $\lambda \in U\subset \mathbb {C}^{n}$. We study the distribution of zeros of $\{f_{\lambda }\}$ in a subdomain $R\subset \subset D$ whose boundary is a closed non-singular analytic curve. As an application, we obtain several results about distributions of zeros of families of generalized exponential polynomials and displacement maps related to certain ODE’s.